@ZettelDistraction said:
The Folgezettel ID 21/2a5b19f can be interpreted as the 6-th note of a sequence starting with 21/2a5b19a, which is a comment on an aspect of 21/2a5b19. That's all the local information in the ID, read from right to left. There is some non-local information: the path to the "root" and the "subject" number 21. Some or all of this data can be moved into the note if other IDs are used.

Maybe there is a little more information embedded in the folgezettel ID. 21/2a5b19f is the 6th comment on 21/2a5b19. A quick and cursory scan of 21/2a5b19f would reveal clues about the topic of the sequence21/2a5b19 and trigger a spider-sense about the "root" subject. A quick look at the note might not identify the notes that make up the path, but it likely will be enough to determine rather a more significant time investment is worthwhile.

I like to read several books and articles at once.

I subscribe to the "start many books" and "finish few" reading method. Finishing only those that capture and hold my attention. These days I'm quick to give up on a book that doesn't hlep produce new and novel ideas.

Not to mention subsequent trains. Otherwise, the train of thought could leave the station or get derailed.

Something has gone haywire with the footnotes. Selecting them puts one on a train that has already left the station. @ctietze, it seems the forum software sets the footnote count per thread rather than per post. A number 1 footnote here links to the first number 1 footnote several posts ago?? This is good to know, and I won't be using numerical keys in my footnotes anymore.

Good luck with that! "You underestimate even the foothills that stand in front of you, and never suspect that far above them, hidden by cloud, rise precipices and snow-fields." Stapledon, Olaf. Last and first men. United States, Dover Publications, 2008.

"By moving yourself, you move your mind." "Silence in the Age of Noise" by Erling Kagge 2016

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

Acknowledgments. Thanks to @ctietze, @Sascha and @Will for helpful comments, criticism and corrections, and to @ctietze and @Sascha for cultivating the zettelkasten.de community and for hosting the forum that motivated—and now hosts—this note.

Introduction

The sociologist Niklas Luhmann assigned unique, immutable identifiers (IDs) to notes (Zettels) within his Zettelkasten to maintain, within the linear ordering of the Zettelkasten, a tree structure that reflected semantic relationships among nearby notes, and that possessed an internal branching property. Luhmann's IDs, sometimes referred to as Folgezettel, were designed to support a researcher who maintains a linear collection of notes, but who works discursively, judiciously taking notes on whatever reading or thinking they happen to be doing at the time, with a view toward future publication—or at minimum, keeping in mind the future self who will be reading them. Each new note is assigned an ID indicating one of the several possible places in the collection where the note either continues a line of thought in a preceding note; comments on or raises a question about some aspect of a preceding note; or starts a new topic.

The new note then inserted into the place in the collection indicated by its ID. The integrity of the collection is maintained by ensuring that related notes are reachable via ID references within notes, either directly or through intermediate sequences of such references. A keyword index is also maintained for the collection. This bottom-up design is intended to enable the researcher to reconstruct their train of thought and resume where they left off, or to follow alternatives chains of notes through the collection, away from the original sequences. In time the collection will have amassed sufficiently many locally-linked and interconnected notes, enough to draft articles, papers or books that are, in some sense latent in the collection, and for which the initial writing has already been done.

Here we will focus on a mathematical formalization of Niklas Luhmann's unique, immutable Zettel IDs. The partially ordered set of generalized Folgezettel IDs is defined first, and then shown to specialize to Luhmann's IDs, up to renaming.

Generalized Folgezettel IDs are coordinates of the nodes of an outline-like tree structure that may branch into any number of other such trees at any node or descendant node. The generalized Folgezettel IDs have the form $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$, where the $(v_i)$ are decimals, i.e., $(j)$-tuples of natural numbers written in the form $(n_1\mathbf{.}n_2\mathbf{.}\ldots\mathbf{.} n_{j})$. Such decimals are familiar as the section and subsection numbers of numbered outlines.

We define an order-preserving bijection from the partially-ordered set of positive Folgezettel IDs (generalized Folgezettel IDs in which all coordinates are positive), to the lexicographically ordered set of normalized decimals, which are decimals in which the initial and final numbers are positive. The linearization map defines a linearization of the partially ordered set of positive Folgezettel IDs; this linearization captures and generalizes the internal branching property of Luhmann's Folgezettel. The linearization lifts to a monoid homomorphism on the monoid of words generated by the symbols of the language of the generalized Folgezettel IDs. This yields an efficient substitution algorithm for linearizing the partial order.

Formalization

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 $(\mathcal{D}^+)$.

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is defined for $(u,v\in \mathcal{D}_0)$ by

$$(u \prec v \Leftrightarrow \begin{cases} \exists x,y,x\in\mathcal{D}_0\cup\left\lbrace\varepsilon\right\rbrace,m,n\in\mathbb{N}, \\\quad\left(u=x\mathbf{.}m\mathbf{.}y\right) \land \left(v=x\mathbf{.}n\mathbf{.}z\right) \land \left(m\lt n\right); \\ \exists x\in\mathcal{D}_0, v = u\mathbf{.}x. \end{cases})$$

Note that in the second alternative above, $(x\ne\varepsilon.)$ The order $(\prec)$ is a lexicographic order, as is the associated partial order $(\preceq)$ defined by $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Each of the structures $$(\left(\mathcal{D}_0,\preceq\right),
\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),
\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),
\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ is lexicographically ordered by a sub-ordering of $(\preceq)$ restricted to the respective domain. We denote any of the lexicographic orders by $(\preceq)$, or by $(\preceq_\mathcal{D}, \preceq_{\mathcal{D}^+})$ as needed.

The nonzero condition rules out zero and "infinitesimals," which are decimals (except for zero) in which every digit is zero.

Note: To establish the linearization of the Folgezettel IDs in the sequel, we will be concerned with the positive and normalized decimals $(\mathcal{D}^+)$ and $(\mathcal{D},)$ respectively.

A generalized Folgezettel ID (or simply, a 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 $(v_1,\ldots, v_k\in\mathcal{D}^+)$ are positive. Folgezettel IDs with nonzero and normalized decimals will serve as counterexamples to the linearization given in the sequel.

The set of Folgezettel IDs is denoted by $(\mathcal{F})$. Observe that for $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k\in\mathcal{F}, )$ $(w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m \in\mathcal{F})$,

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 set of Folgezettel IDs is then $(\mathcal{F}=\mathcal{F}(\mathcal{D}^+).)$ From now on we refer to elements of $(\mathcal{F})$ as IDs.

Luhmann IDs

Notation: write $(`/\text{'})$ for the word $(`\left.\right|_1\text{'})$. An ID $(w\in\mathcal{F})$ is Luhmann-like if $$(w= m\mathbf{.}n / d_1 /\cdots / d_k,)$$ where $(\exists m, n, k, \in\mathbb{Z}^+, d_j\in\mathbb{Z}^+, 1\le j\le k.)$ The Luhmann-like IDs admit a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.

Translation from Luhmann's Folgezettel to Luhmann-like IDs

The translation to and from Luhmann's Folgezettel to Luhmann-like IDs will be given by example. Consider the Folgezettel ID $(21/2a5b19f)$. The corresponding Luhmann-like ID is $$(
21 \mathbf{.} 2 / 1 /5 / 2 /19 /6.)$$ This ID is obtained by replacing the slash "/" with a period, and from the following alternating sequence of letters and numbers by inserting a slash "/" between each contiguous sequence of numbers (letters), and by replacing a letter (or letter sequence, such as "aa", which follows "z") by its corresponding ordinal value in lexicographic order. Thus, $(21/2a5b19f)$ becomes the expression $(21\mathbf{.}2/a/5/b/19/f)$, which then becomes $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$.

The interpretation of $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$ (and hence of $(21/2a5b19f)$) is the 6-th note in a sequence of notes starting with the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$, which itself comments on (or raises a question about) the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19)$. In the notation developed here, the note with decimal ID $(21\mathbf{.}2)$ represents the second note of a sequence starting with $(21\mathbf{.}1)$, under a category (or section) numbered $(21)$.

The reverse translation of a Luhmann-like note ID, such as $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6)$ (or without the syntactic sugar, $(21 \mathbf{.}2 \left.\right|_1 1 \left.\right|_1 5 \left.\right|_1 2 \left.\right|_1 19 \left.\right|_1 6)$), is the reverse process, in which the two-place decimal is replaced with the first number, a slash and the second number. The following slashes are removed in order, where letter sequences replacing numbers alternate with the next adjacent number, and so on until the end. Diagramatically,

$$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6 \rightarrow 21 / 2 a /5 / 2 /19 / 6\rightarrow 21 / 2 a 5 b /19 / 6\rightarrow 21 / 2 a 5 b 19 f)$$

Other embeddings of Luhmann's Folgezettel within the generalized IDs are possible.

Partial order on IDs

The IDs have the structure of a partially ordered set $(\left(\mathcal{F}, \preceq_\mathcal{F}\right))$, where the partial order $(\preceq_\mathcal{F})$ on $(\mathcal{F})$ extends the relation $(\preceq_{\mathcal{D}^+})$ on $(\mathcal{D}^+)$, as follows.

For $(x, y\in\mathcal{F},)$ define $(x\prec_{\mathcal{F}} y)$ by induction on $(y)$.

The four cases of the preceding definition correspond respectively to
1- the comparison of (positive) decimals;
2- the comparison of an ID with a child (i.e., an immediate descendant of the ID);
3- the comparison of IDs on the same branch (i.e., they share the same initial segment ID);
4- the comparison of an ID with a remote descendant.

An inductive argument shows that these cases imply that $(\prec_{\mathcal{F}})$ is transitive.

Fact. If $(x\in\mathcal{F})$, $(y\in\mathcal{D}^+)$ and $(x\prec_{\mathcal{F}^+} y)$, then $(x\in\mathcal{D}^+)$. The proof is immediate, since $(x\in\mathcal{D}^+)$ is the only applicable case.

Linearization

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

Proof. The map $(L)$ is inductively defined by $$(L(w) =
\begin{cases}
w,& w\in\mathcal{D}^+;\\
L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d, &\exists x\in\mathcal{F}^+, n\in\mathbb{Z}^+,d\in\mathcal{D}^+,
w= x\left.\right|_n d.
\end{cases} )$$ for $(w\in \mathcal{F})$.

$(L)$ is well defined.

The map $(L)$ is well-defined (meaning that $(L)$ takes values in the indicated codomain): if $(w\in\mathcal{D}^+)$ then $(L(w)\in\mathcal{D})$. Let $(w\in\mathcal{F})$ and make the induction hypothesis that $$(\forall x\in\mathcal{F}, x\prec w\Rightarrow L(x)\in\mathcal{D}.)$$ Now suppose that the second case of the definition of $(L)$ holds, with $(w= x\left.\right|_n d)$, where $(x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D}.)$ Then $(x\prec w)$ (another induction, by definition of $(\prec)$) and by the induction hypothesis, $(L(x))$ is a normalized decimal; in particular, its initial word is nonzero. Since $(d)$ is a positive (hence normalized) decimal, its final word is nonzero, and therefore the value $$(L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d)$$ of $(L(w))$ is also a normalized decimal.

$(L)$ is injective

To show that $(L)$ is injective, there are three cases for $(v,w\in\mathcal{F})$.
Case 1. $(v,w\in\mathcal{D}^+.)$
Case 2. $(v\in\mathcal{D}^+, w = x\left.\right|_n d)$ for $(x\in\mathcal{F},n\in\mathbb{Z}^+, \in\mathcal{D}^+.)$
Case 3. $(v = y\left.\right|_m c, w = z\left.\right|_n d)$, for $(y,z\in\mathcal{F},m,n\in\mathbb{Z}^+, c,d\in\mathcal{D}^+.)$

Make the induction hypothesis that for $(v,w\in\mathcal{F})$, $(L(v) = L(w) \Rightarrow v = w)$ by induction on $(w)$.

In Case 1, $(L(v) = L(w)\Rightarrow v = w)$ is immediate, since $(L)$ is the identity on $(\mathcal{D}^+)$.

In Case 2, suppose that $(L(v) = L(w))$. By definition of $(L)$,
$$(v=L(x\left.\right|_n d) = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d,)$$ which is impossible since $(v\in\mathcal{D}^+)$ ($(v)$ is a positive decimal) and $(L(w)\in \mathcal{D}\setminus\mathcal{D}^+)$ ($(L(w))$ is a normalized non-positive decimal).

In Case 3, suppose that $(L(y\left.\right|_m c) = L(z\left.\right|_n d).)$ Then, wait for it,
$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}} c =
L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d.
)$$ Now we reason in $(\Sigma^*)$. Since $(c, d)$ are positive decimals, $(c = d)$, so they can be cancelled, leaving$$(
L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}}=
L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}.
)$$ Since $(L(y), L(z))$ are normalized decimals (by the induction above), their final words are nonzero, so that $(m=n.)$ It follows that $(L(y) = L(z))$ and by the induction hypothesis, $(y = z)$. This yields injectivity in Case 3: $$( v = y\left.\right|_m c = z\left.\right|_n d = w)$$.

$(L)$ is surjective.

To show that $(L)$ is surjective, there are two cases, the first being $(c\in\mathcal{D}^+)$, which is immediate ($(L(c)=c)$). If $(c\in\mathcal{D}\setminus\mathcal{D}^+)$, then $(c)$ is normalized but not positive, there is a rightmost consecutive string $(s)$ of zeros of length $(n=|s|\in\mathbb{Z}^+)$ between the initial and final words of $(c)$. This means we can write $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d,)$$ where $(b)$ is a normalized decimal and where $(d)$ is a positive decimal. Make the induction hypothesis on normalized decimals $(c)$, which is that surjectivity holds whenever $(b\in\mathcal{D})$ with $(b\prec c)$, and in that case $(b = L(x))$ for some $(x\in\mathcal{F})$. Then by definition of the map $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x\left.\right|_n d),)$$ where $(x\left.\right|_n d\in\mathcal{F})$. So $(L)$ is surjective.

$(L)$ is order-preserving.

Finally, to show that $(L)$ is order-preserving, let $(x,y\in\mathcal{F})$ with $(x\prec y)$. We proceed by induction on $(y)$. The induction hypothesis (with parameter $(x\in\mathcal{F})$) is that for $(z\in\mathcal{F})$ with $(z\prec y)$, $$(
x\prec z \Rightarrow L(x)\prec L(z).
)$$ There are three cases.

Case 1. The case of $(y\in\mathcal{D}^+, x\prec y)$ forces $(x\in \mathcal{D}^+)$, which yields $(L(x)=x\prec y=L(y))$.

Case 2. If $(y = x |_n d)$ for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then $(x\prec x|_n d)$ so that $$(L(x) \prec L(x) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x |_n d) = L(y). )$$

Case 3. If $(y = z |_n d)$ for some $(z\in\mathcal{F})$ with $(x\prec z)$, and for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then making use of the inductive definition of $(\prec)$ (meaning $(\prec_{\mathcal{F}})$) and the induction hypothesis, $$(L(x) \prec L(z) \prec L(z)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(z|_n d) = L(y). )$$ QED

Linear order from linearization

The linearization map $(L)$ is used to define the linear ordering on the notes of the Zettelkasten. Denote the new ordering on $(\mathcal{F(\mathcal{D}^+)})$ by $(\lll)$. Then the linear ordering is given by
$$(
w\lll x \Leftrightarrow L(w) \prec L(x)
)$$ for $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

The map $(L)$ extends to a monoid homomorphism $(\Sigma^*\rightarrow\Sigma^*)$

The map $(L)$ can be obtained from the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols.
$$(L(w) =
\begin{cases}
w& \text{if}\quad w\in\mathbb{Z}^+;\\
\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&
\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\
\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.
\end{cases} )$$ for $(w\in \Sigma)$. The string substitution is $(O(n))$ in the length $(|n|)$ of the ID.

Counterexamples

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

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

However, the extension of the map $(L)$ to $(\mathcal{G})$ is not injective. Similarly for the extension of the map $(L)$ to the normalized IDs.

A few paragraphs of comments about Folgezttel Formalized in this note were obliterated by the accidental selection and replacement of all of the text, instead of a small portion. This is easy to do on an iPad. Before I could revert to the previous draft, auto save kicked in. I know I could edit offline and upload, but be grateful that I didn't.

The main point: illustrations (preferably programmed in LaTeX) and an interpretation of the notation must be included. There were other remarks about the last minute change of the interpretation of the Folgezttel 21/2a5b19f in the general IDs, but now that will have to wait. The introduction could be edited, references added. The use of Markdown comment syntax in callouts isn't optimal: they scroll horizontally if the text length exceeds the column width. Not a terrific idea to assume that readers will know this. Use something else or get rid of them.

Auto save is forcing practice in concision and is reminding me that the reader's time is valuable.

@ZettelDistraction said:
The Folgezettel ID 21/2a5b19f can be interpreted as the 6-th note of a sequence starting with 21/2a5b19a, which is a comment on an aspect of 21/2a5b19. That's all the local information in the ID, read from right to left. There is some non-local information: the path to the "root" and the "subject" number 21. Some or all of this data can be moved into the note if other IDs are used.

Maybe there is a little more information embedded in the folgezettel ID. 21/2a5b19f is the 6th comment on 21/2a5b19.

As Hakuin is reported to have said, "is that so?" 21/2a5b19f the 6th comment in a sequence of successive continuations of notes, starting with 21/2a5b19a, which is a comment on an aspect of 21/2a5b19, but is it necessarily the case that every member of the sequence comments on an aspect of 21/2a5b19? I am reluctant to commit to this conclusion. Nothing hangs on it as far as the general IDs go. Of course it can be stipulated to hold for some ZK.

Something has gone haywire with the footnotes. Selecting them puts one on a train that has already left the station. @ctietze, it seems the forum software sets the footnote count per thread rather than per post. A number 1 footnote here links to the first number 1 footnote several posts ago?? This is good to know, and I won't be using numerical keys in my footnotes anymore.

Re: footnotes -- the forum is "dumb" in that regard. Your footnote itself becomes the link. So if two or more posts are reachable via the same link, i.e. footnote "1", the target furthest at the top wins. That's how anchor links in HTML work. It'd be more useful if the forum applied some postprocessing so that each footnote is unique, but that's not the case. It also happened on our blog some years ago -- you wouldn't notice the problem when you read 1 article, because there, footnotes are unique on the page, but in an overview of multiple posts, whoops

A formalization and generalization is given of the system of Zettel identifiers (IDs) developed by the sociologist Niklas Luhmann. Luhmann's system of IDs, sometimes called Folgezttel, enabled Luhmann to maintain a dynamically branching tree structure within the linear ordering of the notes of his Zettelkasten. Each note added to the collection is assigned an ID indicating a place in the collection where the note either continues a line of thought in a preceding note; comments on or raises a question about some aspect of a preceding note; or starts a new topic. Further details on Luhmann's Zettelkasten and his system of IDs are given in (Schmidt, J. 2016, 2018).

The partially ordered set of generalized Folgezettel IDs is defined and is shown to specialize to Luhmann's Folgezttel IDs, up to renaming. Generalized Folgezettel IDs are coordinates of the nodes of an outline-like tree structure that may branch into any number of other such trees at any node or descendant node. The generalized Folgezettel IDs have the form $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$, where the $(v_i)$ are decimals, i.e., $(j)$-tuples of natural numbers written in the form $(n_1\mathbf{.}n_2\mathbf{.}\ldots\mathbf{.} n_{j})$. Such decimals are familiar as the section and subsection numbers of numbered outlines.

An order-preserving bijection is defined from the partially-ordered set of positive Folgezettel IDs (generalized Folgezettel IDs in which all coordinates are positive), to the lexicographically ordered set of normalized decimals, which are decimals in which the initial and final numbers are positive. This linearization map is used to define a linearization of the partially ordered set of positive Folgezettel IDs. The linearization generalizes the internal branching property of Luhmann's Folgezettel. The linearization map lifts to a monoid homomorphism on the monoid of words generated by the symbols of the language of the generalized Folgezettel IDs. This yields an efficient substitution algorithm for linearizing the partial order.

Formalization

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 $(\mathcal{D}^+)$.

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is defined for $(u,v\in \mathcal{D}_0)$ by $$(u \prec v \Leftrightarrow \begin{cases} \exists x,y,x\in\mathcal{D}_0\cup\left\lbrace\varepsilon\right\rbrace,m,n\in\mathbb{N}, \\\quad\left(u=x\mathbf{.}m\mathbf{.}y\right) \land \left(v=x\mathbf{.}n\mathbf{.}z\right) \land \left(m\lt n\right); \\ \exists x\in\mathcal{D}_0, v = u\mathbf{.}x. \end{cases})$$

Note that in the second alternative above, $(x\ne\varepsilon.)$ The order $(\prec)$ is a lexicographic order, as is the associated partial order $(\preceq)$ defined by $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Each of the structures $$(\left(\mathcal{D}_0,\preceq\right),
\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),
\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),
\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ is lexicographically ordered by a sub-ordering of $(\preceq)$ restricted to the respective domain. We denote any of the lexicographic orders by $(\preceq)$, or by $(\preceq_\mathcal{D}, \preceq_{\mathcal{D}^+})$ as needed.

The nonzero condition rules out zero and "infinitesimals," which are decimals (except for zero) in which every digit is zero.

Note: The linearization of the Folgezettel IDs in the sequel will rely on the positive and normalized decimals $(\mathcal{D}^+)$ and $(\mathcal{D},)$ respectively.

A generalized Folgezettel ID (or simply, a 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 $(v_1,\ldots, v_k\in\mathcal{D}^+)$ are positive. Folgezettel IDs with nonzero and normalized decimals will serve as counterexamples to the linearization given in the sequel.

The set of Folgezettel IDs is denoted by $(\mathcal{F})$. Observe that for $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k\in\mathcal{F}, )$ $(w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m \in\mathcal{F})$, $$(\left(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k = w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m\right) \Leftrightarrow \left(k=m \land i_n =j_n\, \land v_n=w_n\right))$$ for $(1\le n\le m\text{.})$

Inductive definition of Folgezettel IDs

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 set of Folgezettel IDs is then $(\mathcal{F}=\mathcal{F}(\mathcal{D}^+).)$ From now on elements of $(\mathcal{F})$ will be referred to as IDs.

Luhmann IDs

Notation: write $(`/\text{'})$ for the word $(`\left.\right|_1\text{'})$. An ID $(w\in\mathcal{F})$ is Luhmann-like if $$(w= m\mathbf{.}n / d_1 /\cdots / d_k,)$$ where $(\exists m, n, k, \in\mathbb{Z}^+, d_j\in\mathbb{Z}^+, 1\le j\le k.)$ The Luhmann-like IDs admit a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.

Translation from Luhmann's Folgezettel to Luhmann-like IDs

The translation to and from Luhmann's Folgezettel to Luhmann-like IDs will be given by example. Consider the Folgezettel ID $(21/2a5b19f)$. The corresponding Luhmann-like ID is $$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6.)$$ This ID is obtained by replacing the slash "/" with a period, and from the following alternating sequence of letters and numbers by inserting a slash "/" between each contiguous sequence of numbers (letters), and by replacing a letter (or letter sequence, such as "aa", which follows "z") by its corresponding ordinal value in lexicographic order. Thus, $(21/2a5b19f)$ becomes the expression $(21\mathbf{.}2/a/5/b/19/f)$, which then becomes $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$.

The interpretation of $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$ (and hence of $(21/2a5b19f)$) is the 6-th note in a sequence of notes starting with the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$, which itself comments on (or raises a question about) the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19)$. In the notation developed here, the note with decimal ID $(21\mathbf{.}2)$ represents the second note of a sequence starting with $(21\mathbf{.}1)$, under a category (or section) numbered $(21)$.

The reverse translation of a Luhmann-like note ID, such as $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6)$ (or without the syntactic sugar, $(21 \mathbf{.}2 \left.\right|_1 1 \left.\right|_1 5 \left.\right|_1 2 \left.\right|_1 19 \left.\right|_1 6)$), is the reverse process, in which the two-place decimal is replaced with the first number, a slash and the second number. The following slashes are removed in order, where letter sequences replacing numbers alternate with the next adjacent number, and so on until the end. Diagramatically,
$$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6 \rightarrow 21 / 2 a /5 / 2 /19 / 6\rightarrow 21 / 2 a 5 b /19 / 6\rightarrow 21 / 2 a 5 b 19 f)$$

Other embeddings of Luhmann's Folgezettel within the generalized IDs are possible.

Partial order on IDs

The IDs have the structure of a partially ordered set $(\left(\mathcal{F}, \preceq_\mathcal{F}\right))$, where the partial order $(\preceq_\mathcal{F})$ on $(\mathcal{F})$ extends the relation $(\preceq_{\mathcal{D}^+})$ on $(\mathcal{D}^+)$, as follows.

For $(x, y\in\mathcal{F},)$ define $(x\prec_{\mathcal{F}} y)$ by induction on $(y)$. $$(x\prec_{\mathcal{F}} y\Leftrightarrow \begin{cases} x\prec_{\mathcal{D}^+} y, & x,y\in\mathcal{D}^+;\\ y= x\left.\right|_n d, &\exists n\in\mathbb{Z}^+, d\in\mathcal{D}^+;\\x=z\left.\right|_n c, y= z\left.\right|_n d, &\exists z\in\mathcal{F}, n\in\mathbb{Z}^+,c, d\in\mathcal{D}^+,c\prec_{\mathcal{D}^+}d;\\ y= z\left.\right|_n d,&\exists z\in\mathcal{F}, x\prec_{\mathcal{F}}z, n\in\mathbb{Z}^+, d\in\mathcal{D}^+.\end{cases})$$

The four cases of the preceding definition correspond respectively to
1- the comparison of (positive) decimals;
2- the comparison of an ID with a child (i.e., an immediate descendant of the ID);
3- the comparison of IDs on the same branch (i.e., they share the same initial segment ID);
4- the comparison of an ID with a remote descendant.

An inductive argument shows that these cases imply that $(\prec_{\mathcal{F}})$ is transitive.

Fact. If $(x\in\mathcal{F})$, $(y\in\mathcal{D}^+)$ and $(x\prec_{\mathcal{F}^+} y)$, then $(x\in\mathcal{D}^+)$. The proof is immediate, since $(x\in\mathcal{D}^+)$ is the only applicable case.

Linearization

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

Proof. The map $(L)$ is inductively defined by $$(L(w) =
\begin{cases}
w,& w\in\mathcal{D}^+;\\
L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d, &\exists x\in\mathcal{F}^+, n\in\mathbb{Z}^+,d\in\mathcal{D}^+,
w= x\left.\right|_n d.
\end{cases} )$$ for $(w\in \mathcal{F})$.

$(L)$ is well defined.

The map $(L)$ is well-defined (meaning that $(L)$ takes values in the indicated codomain): if $(w\in\mathcal{D}^+)$ then $(L(w)\in\mathcal{D})$. Let $(w\in\mathcal{F})$ and make the induction hypothesis that $$(\forall x\in\mathcal{F}, x\prec w\Rightarrow L(x)\in\mathcal{D}.)$$ Now suppose that the second case of the definition of $(L)$ holds, with $(w= x\left.\right|_n d)$, where $(x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D}.)$ Then $(x\prec w)$ (another induction, by definition of $(\prec)$) and by the induction hypothesis, $(L(x))$ is a normalized decimal; in particular, its initial word is nonzero. Since $(d)$ is a positive (hence normalized) decimal, its final word is nonzero, and therefore the value $$(L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d)$$ of $(L(w))$ is also a normalized decimal.

$(L)$ is injective

To show that $(L)$ is injective, there are three cases for $(v,w\in\mathcal{F})$.
Case 1. $(v,w\in\mathcal{D}^+.)$
Case 2. $(v\in\mathcal{D}^+, w = x\left.\right|_n d)$ for $(x\in\mathcal{F},n\in\mathbb{Z}^+, \in\mathcal{D}^+.)$
Case 3. $(v = y\left.\right|_m c, w = z\left.\right|_n d)$, for $(y,z\in\mathcal{F},m,n\in\mathbb{Z}^+, c,d\in\mathcal{D}^+.)$

Make the induction hypothesis that for $(v,w\in\mathcal{F})$, $(L(v) = L(w) \Rightarrow v = w)$ by induction on $(w)$.

In Case 1, $(L(v) = L(w)\Rightarrow v = w)$ is immediate, since $(L)$ is the identity on $(\mathcal{D}^+)$.

In Case 2, suppose that $(L(v) = L(w))$. By definition of $(L)$,
$$(v=L(x\left.\right|_n d) = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d,)$$ which is impossible since $(v\in\mathcal{D}^+)$ ($(v)$ is a positive decimal) and $(L(w)\in \mathcal{D}\setminus\mathcal{D}^+)$ ($(L(w))$ is a normalized non-positive decimal).

In Case 3, suppose that $(L(y\left.\right|_m c) = L(z\left.\right|_n d).)$ Then,
$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}} c =
L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d. )$$ Now we reason in $(\Sigma^*)$. Since $(c, d)$ are positive decimals, $(c = d)$, so they can be cancelled, leaving$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}}= L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}. )$$ Since $(L(y), L(z))$ are normalized decimals (by the induction above), their final words are nonzero, so that $(m=n.)$ It follows that $(L(y) = L(z))$ and by the induction hypothesis, $(y = z)$. This yields injectivity in Case 3: $$( v = y\left.\right|_m c = z\left.\right|_n d = w)$$.

$(L)$ is surjective.

To show that $(L)$ is surjective, there are two cases, the first being $(c\in\mathcal{D}^+)$, which is immediate ($(L(c)=c)$). If $(c\in\mathcal{D}\setminus\mathcal{D}^+)$, then $(c)$ is normalized but not positive, there is a rightmost consecutive string $(s)$ of zeros of length $(n=|s|\in\mathbb{Z}^+)$ between the initial and final words of $(c)$. This means we can write $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d,)$$ where $(b)$ is a normalized decimal and where $(d)$ is a positive decimal. Make the induction hypothesis on normalized decimals $(c)$, which is that surjectivity holds whenever $(b\in\mathcal{D})$ with $(b\prec c)$, and in that case $(b = L(x))$ for some $(x\in\mathcal{F})$. Then by definition of the map $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x\left.\right|_n d),)$$ where $(x\left.\right|_n d\in\mathcal{F})$. So $(L)$ is surjective.

$(L)$ is order-preserving.

Finally, to show that $(L)$ is order-preserving, let $(x,y\in\mathcal{F})$ with $(x\prec y)$. We proceed by induction on $(y)$. The induction hypothesis (with parameter $(x\in\mathcal{F})$) is that for $(z\in\mathcal{F})$ with $(z\prec y)$, $$(x\prec z \Rightarrow L(x)\prec L(z).)$$ There are three cases.

Case 1. The case of $(y\in\mathcal{D}^+, x\prec y)$ forces $(x\in \mathcal{D}^+)$, which yields $(L(x)=x\prec y=L(y))$.

Case 2. If $(y = x |_n d)$ for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then $(x\prec x|_n d)$ so that $$(L(x) \prec L(x) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x |_n d) = L(y). )$$

Case 3. If $(y = z |_n d)$ for some $(z\in\mathcal{F})$ with $(x\prec z)$, and for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then making use of the inductive definition of $(\prec)$ (meaning $(\prec_{\mathcal{F}})$) and the induction hypothesis, $$(L(x) \prec L(z) \prec L(z)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(z|_n d) = L(y). )$$ QED

Linear order from linearization

The linearization map $(L)$ is used to define the linear ordering on the notes of the Zettelkasten. Denote the new ordering on $(\mathcal{F(\mathcal{D}^+)})$ by $(\lll)$. Then the linear ordering is given by$$(w\lll x \Leftrightarrow L(w) \prec L(x) )$$ for $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

The map $(L)$ extends to a monoid homomorphism $(\Sigma^*\rightarrow\Sigma^*)$

The map $(L)$ can be obtained from the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols. $$(L(w) =\begin{cases}
w& \text{if}\quad w\in\mathbb{Z}^+;\\
\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&
\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\
\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.
\end{cases} )$$ for $(w\in \Sigma)$. The string substitution is $(O(n))$ in the length $(|n|)$ of the ID.

Counterexamples

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

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

However, the extension of the map $(L)$ to $(\mathcal{G})$ is not injective. Similarly for the extension of the map $(L)$ to the normalized IDs.

Acknowledgments. Thanks to @ctietze, @Sascha and @Will for helpful comments, criticism and corrections.

References

Schmidt, J. (2016). Niklas Luhmann‘s Card Index: Thinking Tool, Communication Partner, Publication Machine. In A. Cevolini (Ed.), Library of the written word: Vol. 53. Forgetting Machines. Knowledge Management Evolution in Early Modern Europe (1. Auflage., pp. 289-311). Leiden: Brill.

After reading "Writing Philosophy" by Richard A. Watson, which was recommended in comments to a post in the Leiter Reports blog, I wrote another draft of Folgezettel Formalized While this isn't philosophy and I am not a philosopher; the math is trivial enough to meet professional standards for mathematical depth within philosophy; Luhmann's theoretical sociology tends to be viewed as philosophy within the Anglophone community of scholars (not to rain on anyone's parade—there are rebuttals in the literature to the claim that Luhmann's work is not even wrong, but none that I am aware of venture to bring Luhmann's abstractions into contact with the Earth—though someone might succeed, God willing); and I thought Prof Watson book was applicable, and pretty well written.

I did take issue with Watson's recommendation to consult the linguistically incompetent Elements of Style by Strunk and White, when the competent and vastly superior Style: Lessons in Clarity and Grace by Joseph M. Williams is available. Also, I don't insist on avoiding the singular 'they', and I am prepared to haul out the 1000+ page descriptivist Cambridge Grammar of the English Language by Rodney Huddleston and Geoffrey Pullum in the almost-lethal grammatical struggle for survival against all fustian prescriptivist adversaries. Unless it means not getting published. Then I will appear to be a completely different person. Following Richard A. Watson's advice to avoid jokes, I decided that the post could convulse its readers into paroxysms of inextinguishable laughter by itself, without my efforts at humor.

Now I am abandoning the LaTeX package xyPic for diagrams in TikZ. This is easy, since xyPic wasn't memorable. Ghosted, after a couple of decades of loyal LaTeX package inclusion, just like that. I will attempt to develop TikZ diagrams for the next draft.

Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. In addition to specifying the location of a Zettel on the spanning tree, a Folgezettel ID indicates whether that Zettel either: continues a prior Zettel; comments on or raises a question about an aspect of a prior Zettel; or begins a new topic.

This is exactly what Folgezettel IDs are mathematically: coordinates of a skeletal backbone for a Zettelkasten. The description occurred to me while I was rewriting a Zettel on a Zettel format within Zettlr that would use Folgezettel, in addition to timestamp IDs. (At this level and this rate, I have long since perished.)

A possible revision. In general, there is a forest of spanning trees within a Zettelkasten. We should be careful not to assume that the graph of Zettels is connected.

A formalization and generalization is given of the system of Zettel identifiers, sometimes called Folgezettel, developed by the sociologist Niklas Luhmann [add reference here]. Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. The graph of Zettels may have several disconnected components, each with its own distingushed spanning tree. In addition to specifying the location of a Zettel on a spanning tree, a Folgezettel ID indicates whether that Zettel either: continues a prior Zettel; comments on or raises a question about an aspect of a prior Zettel; or begins a new topic. Niklas Luhmann's Zettelkasten and his system of Zettel IDs are described in (Schmidt, J. 2016, 2018).

Add references to this site and to the book by @Sascha. @Sascha is the preceding paragraph correct? At some point we should ask Johannes Schmidt, one of the world's experts on the Luhmann archive, whether this is correct.

12021102321344 A mathcha.io generated TikZ diagram

Here is a diagram to illustrate that general Folgezettel are coordinates of indefinitely nested and parallel outlines. The diagram was generated using the https://mathcha.io online interactive editor, hosted in the Republic of Singapore.

The figure below contains a "top level" outline with sections numbered $(1)$, $(1.1)$ and $(1.2)$. The node $(1.1)$ has two comments, indicated by $(1.1|_1 1)$, indicating the first comment on an aspect of $(1.1)$, and $(1.1|_3 1)$, indicating the third comment on aspect of $(1.1)$. The Zettel with ID $(1.1|_1 1.1 |_1 1)$ is the first comment on an aspect of $(1.1|_1 1.1)$. Likewise, $(1.1|_3 1|_5 1)$ is the fifth comment on $(1.1|_3 1)$, and
$(1.1|_3 1|_5 2.5.3)$ is node $(2.5.3)$ within an outline that begins with $(1.1|_3 1|_5 1)$. To illustrate that the numbering handles forests of trees, there is the purple tree.

The terms of the Folgezettel debate

Not shown are internal links within Zettels. However, the formalization illustrates and clarifies what the Folgezettel ID debate is about. A Zettelkasten developed with Folgezettel IDs has a distinguished spanning tree for each of its connected components (in the absence of other links connecting those trees). The term "distinguished" indicates that a specific spanning tree is named by the Folgezettel IDs— these are present in the diagram. In these terms, the Folgezettel debate concerns whether a distingushed spanning tree "backbone" for a Zettelkasten offers the researcher or writer a significant advantage over a Zettelkasten lacking a distinguished spanning tree.

I'm not taking a position. There may be no benefit (that's the null hypothesis); there might be some benefit; there might be an overwhelming payoff with an immense measurable effect size; the payoff could be worse than no Zettelkasten; one might find that structure notes always beat Folgezettel; or perhaps most people would be better off with an "invertebrate" Zettelkasten. In the absence of a framework to make such judgments, I can't say myself.

Finding spanning trees subject to constraints

That there exist spanning trees in digital Zettelkasten with timestamp IDs (for example) is obvious. What isn't obvious is the computational complexity of finding and labelling spanning trees subject to constraints, such as whether there exists a spanning tree with a note labelling such that each note label reflects the decision that was taken when the note was added to the Zettelkasten.

The general Folgezettel really are general

Restricting to one change of branch, i.e., $(z |_1 d)$ up to renumbering, or in words, only one comment per node, is topologically identical with Luhmann's system. In Luhmann's system, the degree of the nodes of the (undirected) spanning trees defined by Folgezettel is always at most $(3)$ (forgetting internal links--there's a forgetful functor for you, though mathematicians sometimes forget to define what a "forgetful functor" is).

This particular diagram cannot be represented with Luhmann's notation, since two comments on the Zettel with ID $(1.1)$ are shown, namely $(1.1|_1 1)$ and $(1.1|_3 1)$. Numerical invariants are more efficient: the degree (total in- plus out-degree if you wish) of $(1.1)$ and $(1.1|_3 1)$ is $(4)$, which of course is greater than the maximum degree $(3)$ of any ID representable with Luhmann's Folgezettel (again forgetting internal links).

20211024164953 Effective Information in Zettelkasten Networks and the Folgezettel Debate

My Zettel title is somewhat pretentious: I haven't measured diddly in Zettelkasten networks. But I want to imagine a possible future.

Only yesterday I said that I knew of no scientific framework for addressing (much less resolving) the Folgezettel debate. Now I think this question isn't as narrow and specialized as it sounds.

One way of stating the question: does the presence of a distinguished spanning tree (the one named by Folgezettel) offer some measurable benefit to the researcher or writer? How do Zettelkasten organized around structure notes compare?

This is still vague, since the notion of "measurable benefit" hasn't been defined. I haven't said anything about the growth of such networks over time; I haven't attempted to define a measure of semantic content within individual Zettels (nevertheless I have some idea where to look); and I haven't stated a criterion for "organization around structure notes" that could even be workshopped, let alone survive peer review.

But applications of information theory to networks is an active area of research. If I were going to think more seriously about this, today I would start here:

The connectivity of a network contains information about the relationships between nodes, which can denote interactions, associations, or dependencies. We show that this information can be analyzed by measuring the uncertainty (and certainty) contained in paths along nodes and links in a network. Specifically, we derive from first principles a measure known as effective information and describe its behavior in common network models. Networks with higher effective information contain more information in the relationships between nodes. We show how subgraphs of nodes can be grouped into macro-nodes, reducing the size of a network while increasing its effective information (a phenomenon known as causal emergence). We find that informative higher scales are common in simulated and real networks across biological, social, informational, and technological domains. These results show that the emergence of higher scales in networks can be directly assessed and that these higher scales offer a way to create certainty out of uncertainty.

...
I did take issue with Watson's recommendation to consult the linguistically incompetent Elements of Style by Strunk and White, when the competent and vastly superior Style: Lessons in Clarity and Grace by Joseph M. Williams is available. Also, I don't insist on avoiding the singular 'they', and I am prepared to haul out the 1000+ page descriptivist Cambridge Grammar of the English Language by Rodney Huddleston and Geoffrey Pullum in the almost-lethal grammatical struggle for survival against all fustian prescriptivist adversaries. Unless it means not getting published. Then I will appear to be a completely different person. Following Richard A. Watson's advice to avoid jokes, I decided that the post could convulse its readers into paroxysms of inextinguishable laughter by itself, without my efforts at humor.

Fellow grammarians unite! One of the threads that keeps me interested in this conversation is your brain-bewitching and sometimes dumbfounding humor. Just as the discussion gets a bit too technical and I get lost in the equations, you drop a pun or joke like dismounting my backpack after the climb into Amy Lake. A combination of the feeling of relief at the new sense of weightlessness and exhilaration because of the natural surroundings. Would you please continue with the jokes! Maybe the jokes can help flush ideas from your thesis. Watson's not watching.

In Luhmann's system, the degree of the nodes of the (undirected) spanning trees defined by Folgezettel is always at most 3 (forgetting internal links--there's a forgetful functor for you, though mathematicians sometimes forget to define what a "forgetful functor" is).

The "forgetful functor" reminds me of the proverbial curse of knowledge or:

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

@ZettelDistraction said:
But applications of information theory to networks is an active area of research. If I were going to think more seriously about this, today I would start here:

This research looks like it might take us somewhere towards the notion of a "measurable benefit." The goal is to measure and incrementally improve the amount and usefulness of information carried in the connection between zettel. Foglezettel IDs is one way. Another is a structure note title/subtitle scheme called structure note.

Information contained in the connections includes, subsumes, and goes beyond the physicalness of the information in the outline syntax. How? I'm not sure? I'll have to look at this paper closer. Especially section IV-A. Selection of real networks. The graphs, created in python, are eye-candy for a budding data spelunker.

It is common to characterize networks based on structural properties like their degree distribution or clustering, and the study of such properties has been crucial for the growth of Network Science. Yet, there remains a gap in our treatment of the information contained in the relationships between nodes in a network, particularly in networks that have both weighted connections and feedback, which are hallmarks of complex systems ^{1}^{2}.

A. Koseska and P. I. H. Bastiaens, Cell signaling as a cognitive process, The EMBO Journal 36, 568 (2017). ↩︎

F. A. Rodrigues, T. K. D. M. Peron, P. Ji, and J. Kurths, The Kuramoto model in complex networks, Physics Reports 610, 1 (2016). ↩︎

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

@ZettelDistraction said:
But applications of information theory to networks is an active area of research. If I were going to think more seriously about this, today I would start here:

This research looks like it might take us somewhere towards the notion of a "measurable benefit."

Well, yes. That's why I mentioned it. Cost benefit would be even better.

The goal is to measure and incrementally improve the amount and usefulness of information carried in the connection between zettel. Foglezettel IDs is one way. Another is a structure note title/subtitle scheme called structure note.

I would start by computing this measure on existing ZK networks, to see whether it yields anything useful. Luhmann's ZK would be a good test case, again assuming the measure will yield something useful. If the measure returns the same value one as a one node network as it does with Luhmann's ZK (unlikely), then either the measure isn't useful or else the measure doesn't detect what made Luhmann's ZK network useful to Luhmann.

At least before attempting to use it for incremental improvements, I would attempt to get some feeling for the measure with examples.

That's my methodological two cents, having computed the Kullback-Liebler divergence of probability distributions of climate-related data. Ideally one would avoid having a distinguished senior scientist look at your calculations and exclaim, "Where are the units in these graphs? This has no scientific interest." (I've seen this, let's say.)

A formalization and generalization is given of the system of Zettel identifiers, sometimes called Folgezettel, developed by the sociologist Niklas Luhmann. Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. The graph of Zettels may have several disconnected components, each with its own distingushed spanning tree. In addition to specifying the location of a Zettel on a spanning tree, a Folgezettel ID indicates whether that Zettel either: continues a prior Zettel; comments on or raises a question about an aspect of a prior Zettel; or begins a new topic. Niklas Luhmann's Zettelkasten and his system of Zettel IDs are described in (Schmidt, J. 2016, 2018).

The partially ordered set of generalized Folgezettel IDs is defined and is shown to specialize to Luhmann's Folgezttel IDs, up to renaming. Generalized Folgezettel IDs are coordinates of the nodes of an outline-like tree structure that may branch into any number of other such trees at any node or descendant node. The generalized Folgezettel IDs have the form $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$, where the $(v_i)$ are decimals, i.e., $(j)$-tuples of natural numbers written in the form $(n_1\mathbf{.}n_2\mathbf{.}\ldots\mathbf{.} n_{j})$. Such decimals are familiar as the section and subsection numbers of numbered outlines.

An order-preserving bijection is defined from the partially-ordered set of positive Folgezettel IDs (generalized Folgezettel IDs in which all coordinates are positive), to the lexicographically ordered set of normalized decimals, which are decimals in which the initial and final numbers are positive. This linearization map is used to define a linearization of the partially ordered set of positive Folgezettel IDs. The linearization generalizes the internal branching property of Luhmann's Folgezettel. The linearization map lifts to a monoid homomorphism on the monoid of words generated by the symbols of the language of the generalized Folgezettel IDs. This yields an algorithm for linearizing the partial order.

Formalization

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 $(\mathcal{D}^+)$.

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is defined for $(u,v\in \mathcal{D}_0)$ by $$(u \prec v \Leftrightarrow \begin{cases} \exists x,y,x\in\mathcal{D}_0\cup\left\lbrace\varepsilon\right\rbrace,m,n\in\mathbb{N}, \\\quad\left(u=x\mathbf{.}m\mathbf{.}y\right) \land \left(v=x\mathbf{.}n\mathbf{.}z\right) \land \left(m\lt n\right); \\ \exists x\in\mathcal{D}_0, v = u\mathbf{.}x. \end{cases})$$

Note that in the second alternative above, $(x\ne\varepsilon.)$ The order $(\prec)$ is a lexicographic order, as is the associated partial order $(\preceq)$ defined by $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Each of the structures $$(\left(\mathcal{D}_0,\preceq\right),
\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),
\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),
\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ is lexicographically ordered by a sub-ordering of $(\preceq)$ restricted to the respective domain. We denote any of the lexicographic orders by $(\preceq)$, or by $(\preceq_\mathcal{D}, \preceq_{\mathcal{D}^+})$ as needed.

The nonzero condition rules out zero and "infinitesimals," which are decimals (except for zero) in which every digit is zero.

Note: The linearization of the Folgezettel IDs in the sequel will rely on the positive and normalized decimals $(\mathcal{D}^+)$ and $(\mathcal{D},)$ respectively.

A generalized Folgezettel ID (or simply, a Folgezettel ID) is a word of $(\Sigma^*)$ of the orm $$(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$$ where $(k\in\mathbb{Z}^+,i_1,\ldots, i_k\in\mathbb{Z}^+)$ and where the decimals $(v_1,\ldots, v_k\in\mathcal{D}^+)$ are positive. Folgezettel IDs with nonzero and normalized decimals will serve as counterexamples to the linearization given in the sequel.

The set of Folgezettel IDs is denoted by $(\mathcal{F})$. Th following diagram illustrates that Folgezettel IDs are coordinates of the nodes of a forest of outline-like tree structures that may branch into any number of other such trees at any node or descendant node. Internal links between nodes are not shown in this diagram.

The notation shown will be interpreted in terms of the choices made when a new node is added top the forest of trees. The figure above contains a "top level" outline with sections numbered $(1)$, $(1.1)$ and $(1.2)$. The node $(1.1)$ has two comments, indicated by $(1.1|_1 1)$, indicating the first comment on an aspect of $(1.1)$, and $(1.1|_3 1)$, indicating the third comment on aspect of $(1.1)$. The Zettel with ID $(1.1|_1 1.1 |_1 1)$ is the first comment on an aspect of $(1.1|_1 1.1)$. Likewise, $(1.1|_3 1|_5 1)$ is the fifth comment on $(1.1|_3 1)$, and $(1.1|_3 1|_5 2.5.3)$ is node $(2.5.3)$ within an outline that begins with $(1.1|_3 1|_5 1)$. The purple tree illustrates that the numbering handles forests of trees.

Inductive definition of Folgezettel IDs

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 set of Folgezettel IDs is then $(\mathcal{F}=\mathcal{F}(\mathcal{D}^+).)$ From now on elements of $(\mathcal{F})$ will be referred to as IDs.

Luhmann IDs

Notation: write $(`/\text{'})$ for the word $(`\left.\right|_1\text{'})$. An ID $(w\in\mathcal{F})$ is Luhmann-like if $$(w= m\mathbf{.}n / d_1 /\cdots / d_k,)$$ where $(\exists m, n, k, \in\mathbb{Z}^+, d_j\in\mathbb{Z}^+, 1\le j\le k.)$ The Luhmann-like IDs admit a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.

Translation from Luhmann's Folgezettel to Luhmann-like IDs

The translation to and from Luhmann's Folgezettel to Luhmann-like IDs will be given by example. Consider the Folgezettel ID $(21/2a5b19f)$. The corresponding Luhmann-like ID is $$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6.)$$ This ID is obtained by replacing the slash "/" with a period, and from the following alternating sequence of letters and numbers by inserting a slash "/" between each contiguous sequence of numbers (letters), and by replacing a letter (or letter sequence, such as "aa", which follows "z") by its corresponding ordinal value in lexicographic order. Thus, $(21/2a5b19f)$ becomes the expression $(21\mathbf{.}2/a/5/b/19/f)$, which then becomes $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$.

The interpretation of $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$ (and hence of $(21/2a5b19f)$) is the 6-th note in a sequence of notes starting with the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$, which itself comments on (or raises a question about) the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19)$. In the notation developed here, the note with decimal ID $(21\mathbf{.}2)$ represents the second note of a sequence starting with $(21\mathbf{.}1)$, under a category (or section) numbered $(21)$.

The reverse translation of a Luhmann-like note ID, such as $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6)$ (or without the syntactic sugar, $(21 \mathbf{.}2 \left.\right|_1 1 \left.\right|_1 5 \left.\right|_1 2 \left.\right|_1 19 \left.\right|_1 6)$), is the reverse process, in which the two-place decimal is replaced with the first number, a slash and the second number. The following slashes are removed in order, where letter sequences replacing numbers alternate with the next adjacent number, and so on until the end. Diagramatically,
$$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6 \rightarrow 21 / 2 a /5 / 2 /19 / 6\rightarrow 21 / 2 a 5 b /19 / 6\rightarrow 21 / 2 a 5 b 19 f)$$

The general Folgezettel IDs versus Luhmann's Folgezettel

Restricting to one change of branch, i.e., $(z |_1 d)$ up to renumbering, or in words, only one comment per node, is topologically identical with Luhmann's system. In Luhmann's system, the degree of the nodes of the (undirected) spanning trees defined by Folgezettel is always at most $(3)$. A simple degree argument shows that diagram above cannot be represented with Luhmann's notation. The degree (total in- plus out-degree if you wish) of $(1.1)$ and $(1.1|_3 1)$ is $(4)$, which of course is greater than the maximum degree $(3)$ of any ID representable with Luhmann's Folgezettel (again forgetting internal links).

Other embeddings of Luhmann's Folgezettel within the generalized IDs are possible.

Partial order on IDs

The IDs have the structure of a partially ordered set $(\left(\mathcal{F}, \preceq_\mathcal{F}\right))$, where the partial order $(\preceq_\mathcal{F})$ on $(\mathcal{F})$ extends the relation $(\preceq_{\mathcal{D}^+})$ on $(\mathcal{D}^+)$, as follows.

For $(x, y\in\mathcal{F},)$ define $(x\prec_{\mathcal{F}} y)$ by induction on $(y)$. $$(x\prec_{\mathcal{F}} y\Leftrightarrow \begin{cases} x\prec_{\mathcal{D}^+} y, & x,y\in\mathcal{D}^+;\\ y= x\left.\right|_n d, &\exists n\in\mathbb{Z}^+, d\in\mathcal{D}^+;\\x=z\left.\right|_n c, y= z\left.\right|_n d, &\exists z\in\mathcal{F}, n\in\mathbb{Z}^+,c, d\in\mathcal{D}^+,c\prec_{\mathcal{D}^+}d;\\ y= z\left.\right|_n d,&\exists z\in\mathcal{F}, x\prec_{\mathcal{F}}z, n\in\mathbb{Z}^+, d\in\mathcal{D}^+.\end{cases})$$

The four cases of the preceding definition correspond respectively to
1. the comparison of (positive) decimals;
2. the comparison of an ID with a child (i.e., an immediate descendant of the ID);
3. the comparison of IDs on the same branch (i.e., they share the same initial segment ID);
4. the comparison of an ID with a remote descendant.

An inductive argument shows that these cases imply that $(\prec_{\mathcal{F}})$ is transitive.

Fact. If $(x\in\mathcal{F})$, $(y\in\mathcal{D}^+)$ and $(x\prec_{\mathcal{F}^+} y)$, then $(x\in\mathcal{D}^+)$. The proof is immediate, since $(x\in\mathcal{D}^+)$ is the only applicable case.

Linearization

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

Proof. The map $(L)$ is inductively defined by $$(L(w) =
\begin{cases}
w,& w\in\mathcal{D}^+;\\
L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d, &\exists x\in\mathcal{F}^+, n\in\mathbb{Z}^+,d\in\mathcal{D}^+,
w= x\left.\right|_n d.
\end{cases} )$$ for $(w\in \mathcal{F})$.

$(L)$ is well defined.

The map $(L)$ is well-defined (meaning that $(L)$ takes values in the indicated codomain): if $(w\in\mathcal{D}^+)$ then $(L(w)\in\mathcal{D})$. Let $(w\in\mathcal{F})$ and make the induction hypothesis that $$(\forall x\in\mathcal{F}, x\prec w\Rightarrow L(x)\in\mathcal{D}.)$$ Now suppose that the second case of the definition of $(L)$ holds, with $(w= x\left.\right|_n d)$, where $(x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D}.)$ Then $(x\prec w)$ (another induction, by definition of $(\prec)$) and by the induction hypothesis, $(L(x))$ is a normalized decimal; in particular, its initial word is nonzero. Since $(d)$ is a positive (hence normalized) decimal, its final word is nonzero, and therefore the value $$(L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d)$$ of $(L(w))$ is also a normalized decimal.

$(L)$ is injective

To show that $(L)$ is injective, there are three cases for $(v,w\in\mathcal{F})$.
Case 1. $(v,w\in\mathcal{D}^+.)$
Case 2. $(v\in\mathcal{D}^+, w = x\left.\right|_n d)$ for $(x\in\mathcal{F},n\in\mathbb{Z}^+, \in\mathcal{D}^+.)$
Case 3. $(v = y\left.\right|_m c, w = z\left.\right|_n d)$, for $(y,z\in\mathcal{F},m,n\in\mathbb{Z}^+, c,d\in\mathcal{D}^+.)$

Make the induction hypothesis that for $(v,w\in\mathcal{F})$, $(L(v) = L(w) \Rightarrow v = w)$ by induction on $(w)$.

In Case 1, $(L(v) = L(w)\Rightarrow v = w)$ is immediate, since $(L)$ is the identity on $(\mathcal{D}^+)$.

In Case 2, suppose that $(L(v) = L(w))$. By definition of $(L)$,
$$(v=L(x\left.\right|_n d) = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d,)$$ which is impossible since $(v\in\mathcal{D}^+)$ ($(v)$ is a positive decimal) and $(L(w)\in \mathcal{D}\setminus\mathcal{D}^+)$ ($(L(w))$ is a normalized non-positive decimal).

In Case 3, suppose that $(L(y\left.\right|_m c) = L(z\left.\right|_n d).)$ Then,
$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}} c =
L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d. )$$ Now we reason in $(\Sigma^*)$. Since $(c, d)$ are positive decimals, $(c = d)$, so they can be cancelled, leaving$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}}= L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}. )$$ Since $(L(y), L(z))$ are normalized decimals (by the induction above), their final words are nonzero, so that $(m=n.)$ It follows that $(L(y) = L(z))$ and by the induction hypothesis, $(y = z)$. This yields injectivity in Case 3: $$( v = y\left.\right|_m c = z\left.\right|_n d = w)$$.

$(L)$ is surjective.

To show that $(L)$ is surjective, there are two cases, the first being $(c\in\mathcal{D}^+)$, which is immediate ($(L(c)=c)$). If $(c\in\mathcal{D}\setminus\mathcal{D}^+)$, then $(c)$ is normalized but not positive, there is a rightmost consecutive string $(s)$ of zeros of length $(n=|s|\in\mathbb{Z}^+)$ between the initial and final words of $(c)$. This means we can write $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d,)$$ where $(b)$ is a normalized decimal and where $(d)$ is a positive decimal. Make the induction hypothesis on normalized decimals $(c)$, which is that surjectivity holds whenever $(b\in\mathcal{D})$ with $(b\prec c)$, and in that case $(b = L(x))$ for some $(x\in\mathcal{F})$. Then by definition of the map $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x\left.\right|_n d),)$$ where $(x\left.\right|_n d\in\mathcal{F})$. So $(L)$ is surjective.

$(L)$ is order-preserving.

Finally, to show that $(L)$ is order-preserving, let $(x,y\in\mathcal{F})$ with $(x\prec y)$. We proceed by induction on $(y)$. The induction hypothesis (with parameter $(x\in\mathcal{F})$) is that for $(z\in\mathcal{F})$ with $(z\prec y)$, $$(x\prec z \Rightarrow L(x)\prec L(z).)$$ There are three cases.

Case 1. The case of $(y\in\mathcal{D}^+, x\prec y)$ forces $(x\in \mathcal{D}^+)$, which yields $(L(x)=x\prec y=L(y))$.

Case 2. If $(y = x |_n d)$ for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then $(x\prec x|_n d)$ so that $$(L(x) \prec L(x) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x |_n d) = L(y). )$$

Case 3. If $(y = z |_n d)$ for some $(z\in\mathcal{F})$ with $(x\prec z)$, and for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then making use of the inductive definition of $(\prec)$ (meaning $(\prec_{\mathcal{F}})$) and the induction hypothesis, $$(L(x) \prec L(z) \prec L(z)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(z|_n d) = L(y). )$$ QED

Linear order from linearization

The linearization map $(L)$ is used to define the linear ordering on the notes of the Zettelkasten. Denote the new ordering on $(\mathcal{F(\mathcal{D}^+)})$ by $(\lll)$. Then the linear ordering is given by$$(w\lll x \Leftrightarrow L(w) \prec L(x) )$$ for $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

The map $(L)$ extends to a monoid homomorphism

The map $(L)$ can be obtained from the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols. $$(L(w) =\begin{cases}
w& \text{if}\quad w\in\mathbb{Z}^+;\\
\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&
\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\
\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.
\end{cases} )$$ for $(w\in \Sigma)$.

Counterexamples

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

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

However, the extension of the map $(L)$ to $(\mathcal{G})$ is not injective. Similarly for the extension of the map $(L)$ to the normalized IDs.

Remarks on spanning trees and the Folgezettel debate

A Zettelkasten developed with Folgezettel IDs has a distinguished spanning tree for each of its connected components (in the absence of other links connecting those trees). The term "distinguished" indicates that a specific spanning tree is named by the Folgezettel IDs—such spanning trees are present in the above diagram, one for each the two trees of the forest shown. In these terms, the Folgezettel debate concerns whether a distinguished spanning tree "backbone" for a Zettelkasten offers the researcher or writer a significant advantage over a Zettelkasten lacking a distinguished spanning tree.

That there exist spanning trees in digital Zettelkasten with timestamp IDs (for example) is obvious. What isn't obvious is the computational complexity of finding and labelling spanning trees subject to constraints, such as finding a spanning tree with a note labelling such that each note label reflects the decision that was taken when the note was added to the Zettelkasten.

Acknowledgments. Thanks to @ctietze, @Sascha and @Will for helpful comments, criticism and corrections.

References

Schmidt, J. (2016). Niklas Luhmann‘s Card Index: Thinking Tool, Communication Partner, Publication Machine. In A. Cevolini (Ed.), Library of the written word: Vol. 53. Forgetting Machines. Knowledge Management Evolution in Early Modern Europe (1. Auflage., pp. 289-311). Leiden: Brill.

R:You stated that links do not inform about past decisions. Compared to Folgezettel? If yes: Both are vague by their nature as such. This is the reason why I proposed the link context. Verbally, the reason why one should follow the link. Less verbally: A description of the type of connection (direction, used model of connection, assumption of part-whole-relationship, etc.). Even less verbally: This could be formalised, perhaps... by you.

Some indication, enough so that the future self has something to go on, such as, "what on earth was I thinking?" or, "Ah, good, now I see what it is, yes, now I understand what they’re at!" [Samuel Beckett. Endgame. Hamm...]

Zettelkasten are so idiosyncratic that it's difficult to know what to formalize.

I think the issue is that the relationship between notes is determined by the relationship of the knowledge that you connect. The connection is not characterised by the fact that there is a connection or the figuration of links between the notes but because of the specific characteristics if you link to an illustrating anecdote or body of evidence, single study etc.

But in any case, the Folgezettel had a built-in discipline: to assign an ID, Luhmann had to determine whether his note continued an existing note, commented on a note without continuing it, or started a new topic altogether. The Folgezettel reflected this much about a note, at the time it was added.

With a digital ZK, it's easy to assign an ID and forget to ask whether the note has any relationship to any other note. That's how mine started (and still is, largely). And so I thought I could use a checklist to slow down...

Structure notes are the digital replacement for this. I'm a little reluctant to build in these local hubs. Perhaps I am too dense to see that the practice realizes "think globally, link locally," adds the element of surprise--it seems arbitrary, the choice of when and where to add them idiosyncratic, dependent on a level of intuition I don't possess. It could become structure notes all the way down...

It is not structure notes in isolation. It is structure note + the habit of starting note create with creating the link to the new note first.

I think it is an empirical question. I could argue (I don't) that Folgezettel lead to a false sense of accomplishment since you have one "link" garantueed. However, to embedd a new piece of knowledge or transform information into knowledge to be able to embedd is dependent on a bit more.

I wouldn't count so much on the element of suprise. It comes from the ability to connect thoughts over infinite time (theoretically) and not from within a moment.

I have the same trepidation that I think Luhmann expressed when he said he wanted to avoid committing to an up-front classification that would lock him in for decades in advance.

For that reason I wanted to add a list of CONTEXT links, with the first a continuation, the links becoming more remote.

This looks like a dead end...

V:Semantic distance cannot be measured. Correct. It is prescriptive since the distance is not semantic sometimes. If you connect a claim with evidence, each evidence could be asigned a value, at least by using an ordinal scale. (e.g. double-blindness of a study increases its value)

There is a family of formal epistemic logics of explicit evidence, under the heading of Justification Logic.

But this is different from assigning a semantic distance between the claim and its evidence. Since there are operations on evidence in these logics, any such assignment, if it had an interpretation within justification logic, would need to respect those operations..

Another example: I use something I call Lindy-Test (named after the Lindy-Effect.

Any phenomenon is run through a series of falsification/validation stages:

It is present in evolutionary history

It is present in religious texts

It is present in classica texts

It is present in contemporary empirical research

It is present in the platonic world (or theory)

I'm unfamiliar with this. Thank you for bringing it to my attention.

I'm still thinking about the mathematical description of Folgezettel. It needs diagrams, which I kept in my head as I was writing. I don't know whether adding the Folgezettel IDs to the Zettel format would help or not. It's possible to traverse them by searching: if you have one, you know what the next one is, if it exists; you know what a side-note is (a descendant, of which there could be many)--a search on a prefix will give you all of them.

My time machine is in the garage. But if the opportunity to move to Vienna ever presents itself, I would not look back.

It is not structure notes in isolation. It is structure note + the habit of starting note create with creating the link to the new note first.

I see this.

I think it is an empirical question. I could argue (I don't) that Folgezettel lead to a false sense of accomplishment since you have one "link" garantueed. However, to embedd a new piece of knowledge or transform information into knowledge to be able to embedd is dependent on a bit more.

True also for structure notes.

V:Semantic distance cannot be measured.

I hadn't actually written this. There are measures at least between words.

This hasn't exactly aged well. Mostly superseded by 20211030133016 Folgezettel Formalized. I could add a my Zettel format to the thread.

Earlier I mentioned that I would include my Zettel format, obtained after the usual futzing, second guessing, backtracking and compulsive oversharing common among ZK-maniacs. The Folgezettel IDs of Luhmann or of 20211030133016 above add too much friction in a digital ZK to be worth the effort. An adaptation of Folgezettel to digital ZK is proposed below. Unresolved links below refer to my own ZK. They aren't necessary to understand this post. I tend to postpone the use of structure notes to the bitter end. The philosophy is think globally, link locally. Next: a checklist, in the spirit of The Checklist Manifesto.

the keyword RELATED followed by a list of IDs of prior Zettels related to immutableID; see 1.c.2 below.

The meta-variables 'immutableID' and 'title' are indexicals in the meta-language. The preceding sentence is in the meta-meta-language since it quotes meta-linguistic terms. The header of this Zettel is in the object language. For the utility of keeping the object language, meta-language and meta-meta-language separate, see Theory of Graded Consequence by Chakraborty and Dutta.^{2}

1.a. YAML frontmatter: optional. If there are Pandoc-style references in the Zettel body, add the following to the YAML frontmatter header.

---
reference-section-title: References
---

References will appear as the last section of the document in Pandoc output. Note that a SYNTAX heading is included in subsequent sections of the format. Comments are called out in blockquotes, like this one.

1.b. Level 1 header including an immutableID and title.

SYNTAX
# immutableIDtitle

The value of immutableID doesn't change, although title might change.

1.c.1. Keywords in #hashtag format.

SYNTAX
#keyword #example

Keywords should be object tags, not category tags.^{3}

1.c.2. RELATED

SYNTAX
The keyword 'RELATED' followed by a comma-separated list of Zettel IDs such that for each ID in the list, immutableID either:

continues ID;

comments on (or interrogates—I hate that usage!) an aspect of ID; or,

is continued by ID.

The RELATED header section is adaptation of Niklas Luhman's Folgezettel ID system to digital Zettelkasten. Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. In addition to specifying the location of a Zettel in a distinguished spanning tree, a Folgezettel ID indicates whether that Zettel:

continues a prior Zettel;

comments on or raises a question about an aspect of a prior Zettel; or,

begins a new topic.

Body: an atomic note.

Often held to consist solely of a "unit of thought." The measure is subjective.

Links, including [[links to other Zettels]], can go here.

Footnotes and endnotes become [[links to other Zettels]] in the body.

In a footnote or endnote Zettel, the ID immutableID is added to RELATED.

You pulled off a thorough discussion of your example by displaying an example. Not the easiest thing to do, but as you told us what you were doing, you did precisely that. Anyone commenting on your zettel format would only be picking around the edges. I'm referring to that "futzing, second-guessing, backtracking, and compulsive oversharing ... ZK-maniac" @Will.

To my knowledge, there is no standard "unit of thought" maintained at the National Institute of Standards and Technology; the Bureau international des poids et mesures; or elsewhere.

Yea, but ... Google tells me that "units of thought" are symbols, concepts, prototypes, images, muscular responses, and language. These originate in that there squishy grey matter bucket and are not relevant to NIST.

When I've done simple Keyboard Maestro scripting, I found that having a UUID with a different format from links and placing it, along with other stuff in the YAML front matter of each note makes scripting easy. This little trick creates a link that surfaces all the inbound links because each zettel has one and only one. It is an excellent hook for scripting.

Keywords should be object tags, not category tags.

This is a bit unsettled in my thinking. Is #zettelkasting an object or a category? Is #thinking-skills an object or a category? They are more of an activity. I see the value tags add as something that is on a spectrum. Complex objects like the #universe, pointing at one end of the spectrum, is worthless for a tag. In comparison, a discrete category such as #objectophilia is specific enough to be of value in tying together the few possible zettel covering this category. In mathematics, categories and objects go together, but who am I to tell you this?

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

@Will said:
You pulled off a thorough discussion of your example by displaying an example. Not the easiest thing to do, but as you told us what you were doing, you did precisely that.

I owe a lot to this community. The least I could do is complete what I set out to do.

Anyone commenting on your zettel format would only be picking around the edges. I'm referring to that "futzing, second-guessing, backtracking, and compulsive oversharing ... ZK-maniac" @Will.

Thank you. Most of my format survived, but thanks to your remarks, I'm going to revise.

@ZettelDistraction joked: To my knowledge, there is no standard "unit of thought" maintained at the National Institute of Standards and Technology; the Bureau international des poids et mesures; or elsewhere.

Yea, but ... Google tells me that "units of thought" are symbols, concepts, prototypes, images, muscular responses, and language. These originate in that there squishy grey matter bucket and are not relevant to NIST.

After some reflection, my candidate for a "unit of thought" is the schema of Cognitive Load Theory (John Sweller). Thank you for prompting me to recall Cognitive Load Theory (CLT).

As for NIST, perhaps as neural computer architecture advances, and provided Sweller's theory accurately predicts that schemata can be identified within the brain (somehow), in the remote future, the NIST museum will exhibit a brain-like neural architecture computer whose internal state demonstrably "encodes" the schemata of John Sweller's theory.

As @MartinBB reminds us, we don't know how the brain works. Nevertheless, bracketing a stupefying level of ignorance, in the nomenclature of CLT, the ideal ZK would help to:

manage intrinsic cognitive load (you do the best you can);

promote germane cognitive load (this helps with schema formation); and,

extirpate extraneous cognitive load (this interferes with schema formation; e.g., my NIST jokes).

How to put this into practice eludes me at the moment. The Cognitive Load-ites provide examples for educators of the three types of cognitive load (if I keep typing I'll manufacture my own load). For now I'll assume that CLT is true, and remark in the section on the Zettel body of the revision that the Zettel format and content should facilitate 1, 2 and 3 (because I assume this is feasible, worthwhile and better than the alternatives).

UPDATE: the voice of @Sascha in my head reminds me that the body of the Zettel should be written for the future self. It should be framed to get the future self to do what it is supposed to do. That's an additional constraint, or guideline. If we only knew how.

Add references to CLT, @Sascha and Frame It Again: New Tools for Rational Decision-Making by José Luis Bermúdez.

I'll never finish...given the impossibility of anticipating every consequence of the design, after the next revision, more tweaking, theorizing, armchair speculation on human psychology and functorial vomitological barfology with coefficients in metrized navel lint modules over graded earwax co-algebras will have taken us past the point of diminishing returns, long after the train has left the abandoned station and the ostrich has finally taken off, at one minute after midnight on the Doomsday Clock. The plan now is to make these few changes, and if we're lucky, to bootstrap our way into some understanding of what we've done. Not to preclude iterating on the design when some or all of my assumptions turn out to be breathtakingly stupid.

When I've done simple Keyboard Maestro scripting, I found that having a UUID with a different format from links and placing it, along with other stuff in the YAML front matter of each note makes scripting easy. This little trick creates a link that surfaces all the inbound links because each zettel has one and only one. It is an excellent hook for scripting.

I guess Keyboard Maestro doesn't make it easy to parse

UUID: [[theUUID]]

I'm Macless, except for a 2007 iMac, which only runs an older version of The Archive. I haven't messed with Keyboard Maestro. You got me there. I'm mostly submerged in Bolgia Two of the Eighth Circle of Windows 10 Hell, in the iconography of Dante's Inferno.

My format is written for Zettlr. I use minimal YAML. I don't want YAMLing Zettels.

Keywords should be object tags, not category tags.

This is a bit unsettled in my thinking. Is #zettelkasting an object or a category? Is #thinking-skills an object or a category? They are more of an activity. I see the value tags add as something that is on a spectrum. Complex objects like the #universe, pointing at one end of the spectrum, is worthless for a tag. In comparison, a discrete category such as #objectophilia is specific enough to be of value in tying together the few possible zettel covering this category. In mathematics, categories and objects go together, but who am I to tell you this?

Following the 𝖅𝖊𝖙𝖙𝖊𝖑-𝕿𝖗𝖆𝖓𝖘𝖋𝖔𝖗𝖒𝖆𝖙𝖎𝖔𝖓𝖕𝖗𝖎𝖓𝖟𝖎𝖕 to reformulate what you've read in your own words, I read @Sascha to say that a hashtag that is specific to the Zettel in which it appears has greater utility than a hashtag that is less specific to the Zettel in which it appears. On the spectrum from concrete to sublimely abstract, choose cement. I could replace the phrase "Keywords should be object tags, not category tags" with "Keywords should be specific to the content of immutableID."

Apologies for another TL;DR. I'm going to maintain the Zettlr Zettel template on GitHub. After a quick review of Efficiency in learning: evidence-based guidelines to manage cognitive load, it's not immediately obvious that extraneous cognitive load ought to be eliminated in every case (this is relative), though generally it is additive. Neither is it obvious that the sole function of the ZK is the creation of schema (it could be one function) or that "atomic note" should be interpreted as a "large chunk" or a schema. This gets us into instructional design. "Processing" Efficiency in learning: evidence-based guidelines to manage cognitive load in the ZK using the Zettel format is likely to be a good test.

I ran out of time before I could include the reference (Clark, Nguyen, and Sweller 2006) below on Cognitive Load Theory.

Nguyen, F., Clark, R. C., Sweller, J. (2006). Efficiency in Learning: Evidence-Based Guidelines to Manage Cognitive Load. Germany: Wiley.

Once again I've spoken too soon. Some humility is in order. In (Brünken, Plass, and Moreno 2010) there are several effects to consider. The additive load hypothesis states that cognitive load is the sum of intrinsic, germane and extraneous cognitive load. There is the expertise reversal effect. The worked example effect. It's not at all obvious how to relate Zettel design and ZK to CLT.

Brünken, R., Plass, J. L., Moreno, R. (2010). Cognitive Load Theory: Theory and Applications. United Kingdom: Cambridge University Press.

@ZettelDistraction was getting at there is no clear definition of a unit of thought, or what it means for a note to be atomic: To my knowledge, there is no standard "unit of thought" maintained at the National Institute of Standards and Technology; the Bureau international des poids et mesures; or elsewhere.

@Will said: Yea, but ... Google tells me that "units of thought" are symbols, concepts, prototypes, images, muscular responses, and language. These originate in that there squishy grey matter bucket and are not relevant to NIST.

How is that helpful? Does the Google search engine have any understanding of "units of thought"? If a symbol is a unit of thought, do you have examples of "atomic" Zettels that contain a single symbol in the body?

My point was that there is no useful measurable standard. Even going by cognitive load, might count as atomic for a novice might be "sub-atomic" and counterproductive for an expert, leading to the expertise reversal effect if the novice and the expert happen to be the same person at different times.

RELATED [[20211115172141]] Zettel format: revised,
[[20210424152745]] Zettel format: rule of threes,
[[20210424174054]] Zettlr title format #+ImmutableID+Title,
[[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

Revision v2 following (Simpson 2021).

This Zettel format only applies to Zettlr+Pandoc+MikTeX+Zotero+BetterBibTex under Windows 10 and has not been tested with other software.

Filenames have the format timestamp.md in my implementation.

1. Header: in three plus 1 parts

An optional YAML frontmatter header with commands to Zettlr and Pandoc;

a Level 1 (H1) header containing an immutable ID, referred to in this Zettel by immutableID, followed by a title, referred to in this Zettel by title;

the keyword RELATED followed by a list of IDs of prior Zettels related to immutableID; see 1.c.2 below.

The meta-variables ‘immutableID’ and ‘title’ are indexicals in the meta-language. The preceding sentence is in the meta-meta-language since it quotes meta-linguistic terms. The header of this Zettel is in the object language. For the utility of keeping the object language, meta-language and meta-meta-language separate, see (Chakraborty, Dutta, and SpringerLink (Online service) 2019).

1.a YAML frontmatter: optional

---
reference-section-title: References
---

If there are Pandoc-style references in the Zettel body, add the preceding YAML frontmatter header to the beginning of immutableID. References will appear as the last section of the document in Pandoc output.

1.b. An immutableID and title at heading level 1

# immutableIDtitle

The value of immutableID doesn’t change, although title might change. This syntax relies on enabling the Zettlr Preferences → Display setting “If present, use the first heading level 1 instead of the filename.” This will display the IDs and titles of Zettel markdown files in the Zettlr file manager pane. Without this setting, the file manager will only show the Zettlr filenames, which in my implementation are IDs.

Paraphrasing Sascha Fast in (Fast 2018), hashtags should be as specific to immutableID as possible.

1.c.2. RELATED Zettel IDs

The keyword RELATED followed by a comma-separated list of Zettel IDs such that for each ID in the list, immutableID either:

continues ID;

comments on (or raises a question about) an aspect of ID; or,

is continued by ID.

The RELATED header section is adaptation of Niklas Luhman’s Folgezettel ID system to digital Zettelkasten. Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. In addition to specifying the location of a Zettel in a distinguished spanning tree, a Folgezettel ID indicates whether that Zettel: continues a prior Zettel; comments on or raises a question about an aspect of a prior Zettel; or, begins a new topic.

2. Body: an atomic note

Some authors recommend limiting the body to a single “unit of thought.”

Links, including [[links to other Zettels]], can go here.

Footnotes and endnotes become [[links to other Zettels]] in the body.

In a footnote or endnote Zettel, the ID immutableID is added to RELATED.

To my knowledge, there is no standard “unit of thought” maintained at the National Institute of Standards and Technology; the Bureau international des poids et mesures; or elsewhere. What a “unit of thought” could mean is time-, context- and writer- dependent. Perhaps an appropriate “unit of thought” is the schema of Cognitive Load Theory (CLT) (Plass, Moreno, and Brünken 2010). Zettels that follow instructional design principles from CLT should: manage intrinsic cognitive load; stimulate germane cognitive load; and, minimize extraneous cognitive load (Clark, Nguyen, and Sweller 2006). Ahrens encourages Zettel writers to use their own words (Ahrens 2017). Arhens leaves no room for direct quotation in Zettels, even in “Literature Notes.” Literature and Permanent Notes considered harmful: see the comments of @ctietze, @Sascha, and @MartinBB in reply to (anonymous 2021). Hans Georg-Moeller speculates that Luhmann’s Zettelkasten contributed to an “unnecessarily convoluted, poorly structured, highly repetitive” writing style lacking a “clear narrative development” (Moeller 2012, chap. 2). An antidote to unclear writing is Style: Lessons in Clarity and Grace (Williams and Bizup 2017). Avoid The Elements of Style. Sascha Fast recommends writing for your future self (Fast 2021). Cory Doctorow maintains that writing and research are distinct, and recommends writing the placeholder TK inline instead of stopping to research (Doctorow 2009). At this point, Zettel writing could use a checklist (Gawande 2010).

3. Footer: References

Pandoc style references in the body are resolved under References, provided the immutableID contains a Pandoc citation and the YAML header of 1.a.

Chakraborty, Mihir Kumar, Soma Dutta, and SpringerLink (Online service).
2019. Theory of Graded Consequence A General Framework for Logics of
Uncertainty.

Clark, Ruth Colvin, Frank Nguyen, and John Sweller. 2006. Efficiency in learning: evidence-based guidelines to manage cognitive load. Essential resources for training and HR professionals. San Francisco, CA: Pfeiffer, a Wiley imprint.

Williams, Joseph M., and Joseph Bizup. 2017. Style: lessons in clarity and grace. Twelfth edition. Always learning. Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris: Pearson.

@ZettelDistraction said:
I'm going to maintain the Zettlr Zettel template on GitHub.

Great idea. I'd love to see what you come up with.

After a quick review of Efficiency in learning: evidence-based guidelines to manage cognitive load, it's not immediately obvious that extraneous cognitive load ought to be eliminated in every case (this is relative), though generally it is additive. Neither is it obvious that the sole function of the ZK is the creation of schema (it could be one function) or that "atomic note" should be interpreted as a "large chunk" or a schema. This gets us into instructional design.

AND

@ZettelDistraction said:
In (Brünken, Plass, and Moreno 2010) there are several effects to consider. The additive load hypothesis states that cognitive load is the sum of intrinsic, germane and extraneous cognitive load. There is the expertise reversal effect. The worked example effect. It's not at all obvious how to relate Zettel design and ZK to CLT.

Brünken, R., Plass, J. L., Moreno, R. (2010). Cognitive Load Theory: Theory and Applications. United Kingdom: Cambridge University Press.

I'm a neophyte with peripheral experience thinking about the effects personally of cognitive load. This topic, particularly the book Efficiency in Learning: Evidence-Based Guidelines to Manage Cognitive Load, has ignited a blooming drive. I've gotten this ebook, and the TOC looks inviting.

Part I. An Introduction to Efficiency in Learning

Part II. Basic Guidelines for Managing Irrelevant Cognitive Load

Part III. Instructional Guidelines for Imposing Relevant Cognitive Load

Part IV. Tailoring Instruction to Learner Expertise

Part V. Cognitive Load Theory in Perspective

This appears to be written with pedagogy in mind, but it is easy to slip instruction to self into the mix.

"Processing" Efficiency in learning: evidence-based guidelines to manage cognitive load in the ZK using the Zettel format is likely to be a good test.

Great idea! Thanks.

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

Your template is better than great. Thinking through the activity of zettelkasting is both fun and gratifying. Your thinking/writing is much finer detailed, and reasoned than my thinking. I'm a Clod-Luddite stumbling through what little time I have left. By your example, you are setting the bar a little higher and pushing me to rethink.

RELATED [[20211115172141]] Zettel format: revised,
[[20210424152745]] Zettel format: rule of threes,
[[20210424174054]] Zettlr title format #+ImmutableID+Title,
[[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

YMMV, I found over time, using the practice of 'just' listing related zettel, either at the top or bottom of a zettel, became too simplistic. Simplistic in terms of meaning. The list became hard to decipher, mainly if it contained more than a couple of links. Some links in this section were clear about the relationship, and some were not, requiring deeper investigation. Of late, I try to put all my links interstitially within the body of the zettel or use a notation structure like the example in the snippet below, where I add a comment under each link's title UUID`. I found this practice helps my future self. **I'm working on the discipline required to follow through with this. **

- Don't need to be exhaustive. **reader will fill in missing details**
- Richard Wright and Haiku [[201911191000]]
- Like a haiku, leave space for the reader
- On-the-nose Writing [[202107230614]]
- opposite of leaving space for the reader

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

RELATED [[20211115172141]] Zettel format: revised,
[[20210424152745]] Zettel format: rule of threes,
[[20210424174054]] Zettlr title format #+ImmutableID+Title,
[[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

YMMV, I found over time, using the practice of 'just' listing related zettel, either at the top or bottom of a zettel, became too simplistic.

I have mismanaged the cognitive load. I think it's worth looking into CLT. I have too many links.

In the Github version, which I intend to create today, I was going to point out that keywords like RELATED in the template have the meaning spelled out in the template, and not the usual dictionary meaning. However, the tendency not to adhere to definitions is so strong a human impulse (I didn't adhere to it) that I'm going to use a term from graph theory instead: ADJACENT. I could have used FOLGEZETTEL, which would be close to my intention, but I won't. What was my intention? To adapt Folgezettel to digital ZK, with timestamps, for example, in a useful way. Folgezettel worked for Luhmann. I'm not yet ready to throw out the analog baby with the analog bathwater (only to have to change the diapers of a baby android).

The idea was to find a small set of neighboring Zettels-- ones that immutableID directly continues; or that comment on (or raise a question about) an aspect of immutableID. I violated this in my example--I will remove a couple. If I add them to the body, then they have to go into the GitHub repository as example Zettels. By ADJACENT I mean those, and only those (maybe I'll give a little and forward link to the immediate follower).

ADJACENT means those closest to immutableID, not just related.

Too bad, I like RELATED better.

What should go into the GitHub repository?

A README.md with: some guidelines on interpretation, and a CC BY-SA 4 notice.

The Zettel template with a CC BY-SA 4 notice (this site is CC BY-SA 4— @Sascha and @ctietze have my undying loyalty on this alone).

Possibly but not right away: a few Zettels to "seed" the repository, such as working with Zotero

A checklist (I have a prototype but I'm not done yet).

If this is any good, others will fork the repository and it adapt for their software, make fun of my choices, maybe come up with variations worth incorporating...or it will sit there moribund.

Note to self: in support of "Avoid The Elements of Style" in the template remarks (3. Body), add the citation (Pullum, “The Land of the Free and The Elements of Style,” English Today 102, vol. 26, no. 2 [June 2010], introduction.)

The keyword RELATED was replaced with CONTEXT. I added an updated version of the template to the README.md file itself. The original markdown file for the template is slightly behind. This will have to be updated.

A version stripped down to essentials would be better to start with. It's hard to see the simple structure. There are a couple of variations depending whether the references are present, entered by hand or generated with Zotero.

An abbreviated template, with a checklist. The checklist Markdown works better on GitHub at https://github.com/flengyel/Zettel and in Zettlr. A revision of the longer Zettel template follows. @Will Simpson made me realize that the keywordRELATEDam biguous, so I replaced this with the less—or at least "differently"—biguous keywordCONTEXT. Apologies for an unambiguously desperate pun.

A Zettel has three parts: a header, a body, and a footer. The header starts with an optional YAML header and includes a level 1 (H1) header with a timestamp ID and a title; a list of hashtag keywords; and a CONTEXT line that links to a Zettel that this Zettel continues or comments (or raises a question about) on an aspect of that Zettel (if there such a Zettel). This text is part of the body. Footnotes and endnotes become links within the body to other Zettels. That leaves the References section for the footer. The References section below can be omitted, generated with a references manager such as Zotero and a YAML header [[20211118010533]] Zettel template v2.1; or added by hand, as below.

This Zettel format only applies to Zettlr 2.0.3 + Pandoc 2.16.1 + MikTeX 21.2-x64 + Zotero 5.0.96.3 + BetterBibTex 5.6.8 under Windows 10 and has not been tested with other software.

Filenames have the format timestamp.md in my implementation.

Every Zettel is a note. Literature and Permanent notes (Ahrens 2017) considered harmful (anonymous 2021).

The following checklist in Markdown is a first attempt (Gawande 2010).

[ ] Replace the header timestamp ID with your own ID.

[ ] Replace the title with your title.

[ ] Replace the hashtags above with your own #hashtag keywords. You may need to indent your hashtags a space if the first one becomes an (H1) header, or else escape them as follows: \#escapedhashtag.

[ ] Replace the link after the CONTEXT keyword with a link to a Zettel Z such that this Zettel

[ ] continues Z; or

[ ] comments on (or raises a question about) an aspect of Zettel Z; otherwise,

[ ] if there is no such prior Zettel Z, and this Zettel starts a new topic, remove the CONTEXT line entirely.

[ ] Replace the body (including this cheklist) with your own text and Markdown in the body.

[ ] Are there footnotes or endnotes?

[ ] Footnotes and endnotes become links within the body to other Zettels.

[ ] If there are literature citations from a citation manager (Zotero is assumed here)

[ ] Add citations in Pandoc format.

[ ] Verify that the Zotero Citation Database is exported in CSL JSON format and specified in Preferences → Export → Citation Database;

[ ] Add a YAML header to beginning of this document (see [[20211118010533]] Zettel template v2.1); or

[ ] A references section can be added by hand, as below.

Revision v2.1 following remarks by (Simpson 2021), with additional citations.

This Zettel format only applies to Zettlr 2.0.3 + Pandoc 2.16.1 + MikTeX 21.2-x64 + Zotero 5.0.96.3 + BetterBibTex 5.6.8 under Windows 10 and has not been tested with other software.

Filenames have the format timestamp.md in my implementation.

1. Header: in 3 + 1 parts

An optional YAML frontmatter header with commands to Zettlr and Pandoc;

a Level 1 (H1) header containing an immutable ID, referred to in this Zettel by immutableID, followed by a title, referred to in this Zettel by title;

the keyword CONTEXT followed by a list of IDs of prior Zettels providing the immediate context for immutableID; see 1.c.2 below.

The meta-variables ‘immutableID’ and ‘title’ are indexicals in the meta-language. The preceding sentence is in the meta-meta-language since it quotes meta-linguistic terms. The header of this Zettel is in the object language. For the utility of keeping the object language, meta-language and meta-meta-language separate, see (Chakraborty, Dutta, and SpringerLink (Online service) 2019).

1.a YAML frontmatter: optional

---
reference-section-title: References
---

If there are Pandoc-style references in the Zettel body, add the preceding YAML frontmatter header to the beginning of immutableID. References will appear as the last section of the document in Pandoc output.

1.b. An immutableID and title at heading level 1

# immutableIDtitle

[[20210424174054]] Zettlr title format #+ImmutableID+Title,

The value of immutableID doesn’t change, although title might change. This syntax relies on enabling the Zettlr Preferences → Display setting “If present, use the first heading level 1 instead of the filename.” This will display the IDs and titles of Zettel markdown files in the Zettlr file manager pane. Without this setting, the file manager will only show the Zettlr filenames, which in my implementation are IDs.

1.c.1. Keywords in #hashtag format

#keyword #example

Paraphrasing Sascha Fast in (Fast 2018), hashtags should be as specific to immutableID as possible.

1.c.2. CONTEXT Zettel IDs

The keyword CONTEXT followed by a comma-separated list of Zettel IDs such that for each ID in the list, immutableID either:

continues ID;

comments on (or raises a question about) an aspect of ID; or,

is continued by ID.

No other Zettel ID belongs with CONTEXT,

The keyword CONTEXT applies to Zettel IDs satisfying the above conditions only, including those that might provide context for ```immutableID`` in other senses of the term.

The CONTEXT header section is adaptation of Niklas Luhmann’s Folgezettel ID system to digital Zettelkasten. Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. In addition to specifying the location of a Zettel in a distinguished spanning tree, a Folgezettel ID indicates whether that Zettel: continues a prior Zettel; comments on or raises a question about an aspect of a prior Zettel; or, begins a new topic.This last case is not covered under CONTEXT. Under consideration: an additional keyword PROTEXT for IDs for which immutableID provides the CONTEXT.

2. Body: an atomic note

Some authors recommend limiting the body to a single “unit of thought.”

Links [[to other Zettels]] and external links can go here.

Footnotes and endnotes become links [[to other Zettels]] in the body.

In a footnote or endnote Zettel, the ID immutableID is added to RELATED.

To my knowledge, there is no standard “unit of thought” maintained at the National Institute of Standards and Technology; the Bureau international des poids et mesures; or elsewhere. What a “unit of thought” could mean is time-, context- and writer-dependent. Perhaps an appropriate “unit of thought” is the schema of Cognitive Load Theory (CLT) (Plass, Moreno, and Brünken 2010). Zettels that follow instructional design principles from CLT should: manage intrinsic cognitive load; stimulate germane cognitive load; and, minimize extraneous cognitive load (Clark, Nguyen, and Sweller 2006). Ahrens encourages Zettel writers to use their own words (Ahrens 2017). Arhens leaves no room for direct quotation in Zettels, even in “Literature Notes.” Literature and Permanent Notes considered harmful: see the comments of @ctietze, @Sascha, and @MartinBB in reply to (anonymous 2021). Hans Georg-Moeller speculates that Luhmann’s Zettelkasten contributed to an “unnecessarily convoluted, poorly structured, highly repetitive” writing style lacking a “clear narrative development” (Moeller 2012, chap. 2). An antidote to unclear writing is Style: Lessons in Clarity and Grace (Williams and Bizup 2017). Avoid The Elements of Style (Pullum 2010, 2014). Sascha Fast recommends writing for your future self (Fast 2021). Cory Doctorow maintains that writing and research are distinct, and recommends writing the placeholder TK inline instead of stopping to research (Doctorow 2009). At this point, Zettel writing could use a checklist (Gawande 2010).

3. Footer: References

[[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

Pandoc style references in the body are resolved under References, provided the immutableID contains a Pandoc citation and the YAML header of 1.a.

Chakraborty, Mihir K., and Soma Dutta. 2019. Theory of graded consequence: a general framework for logics of uncertainty. Logic in Asia: Studia Logica Library. Singapore: Springer.

Clark, Ruth Colvin, Frank Nguyen, and John Sweller. 2006. Efficiency in learning: evidence-based guidelines to manage cognitive load. Essential resources for training and HR professionals. San Francisco, CA: Pfeiffer, a Wiley imprint.

Williams, Joseph M., and Joseph Bizup. 2017. Style: lessons in clarity and grace. Twelfth edition. Always learning. Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris: Pearson.**

@ZettelDistraction, thanks for sharing your ideas about a checklist. Below is my note on this topic with revisions spurred by your example.

I am thinking in the context of a simplified workflow. I create notes exclusively using the templating the software stack I use allows. It provides an algorithmically based choice for some of the items on the checklist based on the template choice. Because the software handles them, it seems supercilious to have them on the checklist.

What I record for the title, subtitle^{1}, and tags are place-holders initially. They best-worked during, and after the note is drafted. This circular feedback makes the structure/order of the checklist items a bit hard to formalize.

[edit]
My workflow/practice is evolving towards placing tags and links interstitially within the note. This is a more fine-grained approach than placing them all together.
[/edit]

If surgeons and airline pilots can benefit from checklists, they probably will help me.^{2}

It is essential to keep any formal checklist flexible and straightforward.
If every zettel is an exception, it points to a failure of the checklist to provide guidance.
There is a difference between the structure of a zettel and its content.
The checklist guides the structure but only hints/prompts the content.
My idea of a zettel-making-checklist is that it focuses on the structure, thereby freeing up cognitive cycles so I can focus on the zettel content. And heaven knows I need all the cognitive cycles I can muster.

New Note

YAML Frontmatter [[202003231450]]
[ ] UUID present? (Provided by the software preferences.)
[ ] Creation date and time present? (Provided by the software preferences.)
[ ] Tags present? (Added and revised during or after the body is created.)

Body
[ ] Title present? (Reconsider title as note develops. The title wants to be short, concise, and relevant. It wants to be descriptive of the content.)
[ ] Is a one or two summary sentence present at the head of the note. It should answer the question, "What is the central/essential thing to remember?" (Best done after the note is developed in draft one and continually revisited.)

Actual zettel (may include one or more of the following)

Argument

Counterarguments

Factual claim

Reference (to other text)

Quotation (only the pithiest)

Metaphor (a figurative comparison)

Summary of idea

Definition

Question posed by the author

Questioning the author

An idea #2do

[ ] Connect to an existing structure note or make this the first zettel connection in a new structure note?
[ ] Ask if deep links are present? Look for connections with keyword and key idea searches.
[ ] Is all #beautiful language captured and tagged

Footer/References/Citations/NOtes
[ ] Footnotes
[ ] Bibliographic citations as needed?
[ ] Are they present in Zotero?
[ ] URLS (web, Evernote, Bear)

This is only the first draft. Refactor, Refactor, Refactor!

Is a one or two summary sentence present answering the question, What is the central/essential thing to remember? ↩︎

Will Simpson
My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

@Will said: @ZettelDistraction, thanks for sharing your ideas about a checklist. Below is my note on this topic with revisions spurred by your example.
.... This is only the first draft. Refactor, Refactor, Refactor!

[^subtitle]: Is a one or two summary sentence present answering the question, What is the central/essential thing to remember?

Our workflows and formats are different enough that at most I will borrow the idea to add a step at the end to take a second pass at the title, keywords, etc., and perhaps add a summary sentence.

I think my checklist and format attempts to separate concerns...

I could add the formats and checklists you and others use there. It might be helpful to start a repository of Zettel formats, checklists and workflows that different people use. Examples can be helpful.

Your signature has been updated:

"I'm a futzing, second-guessing, backtracking, compulsive oversharing, ZK-maniac, in other words, your typical zettelnant."

## Comments

Maybe there is a little more information embedded in the folgezettel ID.

`21/2a5b19f`

is the 6th comment on`21/2a5b19`

. A quick and cursory scan of`21/2a5b19f`

would reveal clues about the topic of the sequence`21/2a5b19`

and trigger a spider-sense about the "root" subject. A quick look at the note might not identify the notes that make up the path, but it likely will be enough to determine rather a more significant time investment is worthwhile.I subscribe to the "start many books" and "finish few" reading method. Finishing only those that capture and hold my attention. These days I'm quick to give up on a book that doesn't hlep produce new and novel ideas.

Something has gone haywire with the footnotes. Selecting them puts one on a train that has already left the station.

@ctietze, it seems the forum software sets the footnote count per thread rather than per post. A number 1 footnote here links to the first number 1 footnote several posts ago?? This is good to know, and I won't be using numerical keys in my footnotes anymore."By moving yourself, you move your mind."

"Silence in the Age of Noise"by Erling Kagge 2016Will Simpson

My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time.

kestrelcreek.com

## 20211016193154 Folgezettel Formalized

F Lengyel

#folgezettel #niklasluhmann #combinatorics #poset

Draft only: not for distributionJust kidding 笑 (although it is a draft). This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Acknowledgments. Thanks to @ctietze, @Sascha and @Will for helpful comments, criticism and corrections, and to @ctietze and @Sascha for cultivating the zettelkasten.de community and for hosting the forum that motivated—and now hosts—this note.## Introduction

The sociologist Niklas Luhmann assigned unique, immutable identifiers (IDs) to notes (Zettels) within his Zettelkasten to maintain, within the linear ordering of the Zettelkasten, a tree structure that reflected semantic relationships among nearby notes, and that possessed an internal branching property. Luhmann's IDs, sometimes referred to as Folgezettel, were designed to support a researcher who maintains a linear collection of notes, but who works discursively, judiciously taking notes on whatever reading or thinking they happen to be doing at the time, with a view toward future publication—or at minimum, keeping in mind the future self who will be reading them. Each new note is assigned an ID indicating one of the several possible places in the collection where the note either continues a line of thought in a preceding note; comments on or raises a question about some aspect of a preceding note; or starts a new topic.

The new note then inserted into the place in the collection indicated by its ID. The integrity of the collection is maintained by ensuring that related notes are reachable via ID references within notes, either directly or through intermediate sequences of such references. A keyword index is also maintained for the collection. This bottom-up design is intended to enable the researcher to reconstruct their train of thought and resume where they left off, or to follow alternatives chains of notes through the collection, away from the original sequences. In time the collection will have amassed sufficiently many locally-linked and interconnected notes, enough to draft articles, papers or books that are, in some sense latent in the collection, and for which the initial writing has already been done.

Here we will focus on a mathematical formalization of Niklas Luhmann's unique, immutable Zettel IDs. The partially ordered set of generalized Folgezettel IDs is defined first, and then shown to specialize to Luhmann's IDs, up to renaming.

Generalized Folgezettel IDs are coordinates of the nodes of an outline-like tree structure that may branch into any number of other such trees at any node or descendant node. The generalized Folgezettel IDs have the form $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$, where the $(v_i)$ are decimals, i.e., $(j)$-tuples of natural numbers written in the form $(n_1\mathbf{.}n_2\mathbf{.}\ldots\mathbf{.} n_{j})$. Such decimals are familiar as the section and subsection numbers of numbered outlines.

We define an order-preserving bijection from the partially-ordered set of positive Folgezettel IDs (generalized Folgezettel IDs in which all coordinates are positive), to the lexicographically ordered set of normalized decimals, which are decimals in which the initial and final numbers are positive. The linearization map defines a linearization of the partially ordered set of positive Folgezettel IDs; this linearization captures and generalizes the internal branching property of Luhmann's Folgezettel. The linearization lifts to a monoid homomorphism on the monoid of words generated by the symbols of the language of the generalized Folgezettel IDs. This yields an efficient substitution algorithm for linearizing the partial order.

## Formalization

## 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 (

akathe 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

decimalis an element of the set $(\mathcal{D}_0)$. A decimal isnonzeroif at least one of its integer symbols is nonzero; the set of nonzero decimals is denoted $(\mathcal{D}_{\ne0})$. A decimal isnormalizedif its first and last integer symbols are nonzero; the set of normalized decimals is denoted $(\mathcal{D})$. A decimal ispositiveif all of its integer symbols are nonzero; the set of positive decimals is denoted $(\mathcal{D}^+)$.## Examples

## Lexicographic order on decimals

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is defined for $(u,v\in \mathcal{D}_0)$ by

Note that in the second alternative above, $(x\ne\varepsilon.)$ The order $(\prec)$ is a lexicographic order, as is the associated partial order $(\preceq)$ defined by $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Each of the structures $$(\left(\mathcal{D}_0,\preceq\right),

\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),

\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),

\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ is lexicographically ordered by a sub-ordering of $(\preceq)$ restricted to the respective domain. We denote any of the lexicographic orders by $(\preceq)$, or by $(\preceq_\mathcal{D}, \preceq_{\mathcal{D}^+})$ as needed.

## Examples

The nonzero condition rules out zero and "infinitesimals," which are decimals (except for zero) in which every digit is zero.

Note: To establish the linearization of the Folgezettel IDs in the sequel, we will be concerned with the positive and normalized decimals $(\mathcal{D}^+)$ and $(\mathcal{D},)$ respectively.A

generalized Folgezettel ID(or simply, a 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 $(v_1,\ldots, v_k\in\mathcal{D}^+)$ are positive. Folgezettel IDs with nonzero and normalized decimals will serve as counterexamples to the linearization given in the sequel.

The set of Folgezettel IDs is denoted by $(\mathcal{F})$. Observe that for $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k\in\mathcal{F}, )$ $(w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m \in\mathcal{F})$,

## Inductive definition of Folgezettel IDs

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 set of Folgezettel IDs is then $(\mathcal{F}=\mathcal{F}(\mathcal{D}^+).)$ From now on we refer to elements of $(\mathcal{F})$ as IDs.

## Luhmann IDs

Notation: write $(`/\text{'})$ for the word $(`\left.\right|_1\text{'})$. An ID $(w\in\mathcal{F})$ is

Luhmann-likeif $$(w= m\mathbf{.}n / d_1 /\cdots / d_k,)$$ where $(\exists m, n, k, \in\mathbb{Z}^+, d_j\in\mathbb{Z}^+, 1\le j\le k.)$ The Luhmann-like IDs admit a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.## Translation from Luhmann's Folgezettel to Luhmann-like IDs

The translation to and from Luhmann's Folgezettel to Luhmann-like IDs will be given by example. Consider the Folgezettel ID $(21/2a5b19f)$. The corresponding Luhmann-like ID is $$(

21 \mathbf{.} 2 / 1 /5 / 2 /19 /6.)$$ This ID is obtained by replacing the slash "/" with a period, and from the following alternating sequence of letters and numbers by inserting a slash "/" between each contiguous sequence of numbers (letters), and by replacing a letter (or letter sequence, such as "aa", which follows "z") by its corresponding ordinal value in lexicographic order. Thus, $(21/2a5b19f)$ becomes the expression $(21\mathbf{.}2/a/5/b/19/f)$, which then becomes $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$.

The interpretation of $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$ (and hence of $(21/2a5b19f)$) is the 6-th note in a sequence of notes starting with the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$, which itself comments on (or raises a question about) the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19)$. In the notation developed here, the note with decimal ID $(21\mathbf{.}2)$ represents the second note of a sequence starting with $(21\mathbf{.}1)$, under a category (or section) numbered $(21)$.

The reverse translation of a Luhmann-like note ID, such as $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6)$ (or without the syntactic sugar, $(21 \mathbf{.}2 \left.\right|_1 1 \left.\right|_1 5 \left.\right|_1 2 \left.\right|_1 19 \left.\right|_1 6)$), is the reverse process, in which the two-place decimal is replaced with the first number, a slash and the second number. The following slashes are removed in order, where letter sequences replacing numbers alternate with the next adjacent number, and so on until the end. Diagramatically,

Other embeddings of Luhmann's Folgezettel within the generalized IDs are possible.

## Partial order on IDs

The IDs have the structure of a partially ordered set $(\left(\mathcal{F}, \preceq_\mathcal{F}\right))$, where the partial order $(\preceq_\mathcal{F})$ on $(\mathcal{F})$ extends the relation $(\preceq_{\mathcal{D}^+})$ on $(\mathcal{D}^+)$, as follows.

For $(x, y\in\mathcal{F},)$ define $(x\prec_{\mathcal{F}} y)$ by induction on $(y)$.

The four cases of the preceding definition correspond respectively to

1- the comparison of (positive) decimals;

2- the comparison of an ID with a child (i.e., an immediate descendant of the ID);

3- the comparison of IDs on the same branch (i.e., they share the same initial segment ID);

4- the comparison of an ID with a remote descendant.

An inductive argument shows that these cases imply that $(\prec_{\mathcal{F}})$ is transitive.

Fact. If $(x\in\mathcal{F})$, $(y\in\mathcal{D}^+)$ and $(x\prec_{\mathcal{F}^+} y)$, then $(x\in\mathcal{D}^+)$. The proof is immediate, since $(x\in\mathcal{D}^+)$ is the only applicable case.## Linearization

Proof. The map $(L)$ is inductively defined by $$(L(w) =

\begin{cases}

w,& w\in\mathcal{D}^+;\\

L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d, &\exists x\in\mathcal{F}^+, n\in\mathbb{Z}^+,d\in\mathcal{D}^+,

w= x\left.\right|_n d.

\end{cases} )$$ for $(w\in \mathcal{F})$.

## $(L)$ is well defined.

The map $(L)$ is well-defined (meaning that $(L)$ takes values in the indicated codomain): if $(w\in\mathcal{D}^+)$ then $(L(w)\in\mathcal{D})$. Let $(w\in\mathcal{F})$ and make the induction hypothesis that $$(\forall x\in\mathcal{F}, x\prec w\Rightarrow L(x)\in\mathcal{D}.)$$ Now suppose that the second case of the definition of $(L)$ holds, with $(w= x\left.\right|_n d)$, where $(x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D}.)$ Then $(x\prec w)$ (another induction, by definition of $(\prec)$) and by the induction hypothesis, $(L(x))$ is a normalized decimal; in particular, its initial word is nonzero. Since $(d)$ is a positive (hence normalized) decimal, its final word is nonzero, and therefore the value $$(L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d)$$ of $(L(w))$ is also a normalized decimal.

## $(L)$ is injective

To show that $(L)$ is injective, there are three cases for $(v,w\in\mathcal{F})$.

Case 1. $(v,w\in\mathcal{D}^+.)$

Case 2. $(v\in\mathcal{D}^+, w = x\left.\right|_n d)$ for $(x\in\mathcal{F},n\in\mathbb{Z}^+, \in\mathcal{D}^+.)$

Case 3. $(v = y\left.\right|_m c, w = z\left.\right|_n d)$, for $(y,z\in\mathcal{F},m,n\in\mathbb{Z}^+, c,d\in\mathcal{D}^+.)$

Make the induction hypothesis that for $(v,w\in\mathcal{F})$, $(L(v) = L(w) \Rightarrow v = w)$ by induction on $(w)$.

In Case 1, $(L(v) = L(w)\Rightarrow v = w)$ is immediate, since $(L)$ is the identity on $(\mathcal{D}^+)$.

In Case 2, suppose that $(L(v) = L(w))$. By definition of $(L)$,

$$(v=L(x\left.\right|_n d) = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d,)$$ which is impossible since $(v\in\mathcal{D}^+)$ ($(v)$ is a positive decimal) and $(L(w)\in \mathcal{D}\setminus\mathcal{D}^+)$ ($(L(w))$ is a normalized non-positive decimal).

In Case 3, suppose that $(L(y\left.\right|_m c) = L(z\left.\right|_n d).)$ Then, wait for it,

$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}} c =

L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d.

)$$ Now we reason in $(\Sigma^*)$. Since $(c, d)$ are positive decimals, $(c = d)$, so they can be cancelled, leaving$$(

L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}}=

L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}.

)$$ Since $(L(y), L(z))$ are normalized decimals (by the induction above), their final words are nonzero, so that $(m=n.)$ It follows that $(L(y) = L(z))$ and by the induction hypothesis, $(y = z)$. This yields injectivity in Case 3: $$( v = y\left.\right|_m c = z\left.\right|_n d = w)$$.

## $(L)$ is surjective.

To show that $(L)$ is surjective, there are two cases, the first being $(c\in\mathcal{D}^+)$, which is immediate ($(L(c)=c)$). If $(c\in\mathcal{D}\setminus\mathcal{D}^+)$, then $(c)$ is normalized but not positive, there is a rightmost consecutive string $(s)$ of zeros of length $(n=|s|\in\mathbb{Z}^+)$ between the initial and final words of $(c)$. This means we can write $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d,)$$ where $(b)$ is a normalized decimal and where $(d)$ is a positive decimal. Make the induction hypothesis on normalized decimals $(c)$, which is that surjectivity holds whenever $(b\in\mathcal{D})$ with $(b\prec c)$, and in that case $(b = L(x))$ for some $(x\in\mathcal{F})$. Then by definition of the map $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x\left.\right|_n d),)$$ where $(x\left.\right|_n d\in\mathcal{F})$. So $(L)$ is surjective.

## $(L)$ is order-preserving.

Finally, to show that $(L)$ is order-preserving, let $(x,y\in\mathcal{F})$ with $(x\prec y)$. We proceed by induction on $(y)$. The induction hypothesis (with parameter $(x\in\mathcal{F})$) is that for $(z\in\mathcal{F})$ with $(z\prec y)$, $$(

x\prec z \Rightarrow L(x)\prec L(z).

)$$ There are three cases.

Case 1. The case of $(y\in\mathcal{D}^+, x\prec y)$ forces $(x\in \mathcal{D}^+)$, which yields $(L(x)=x\prec y=L(y))$.

Case 2. If $(y = x |_n d)$ for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then $(x\prec x|_n d)$ so that $$(L(x) \prec L(x) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x |_n d) = L(y). )$$

Case 3. If $(y = z |_n d)$ for some $(z\in\mathcal{F})$ with $(x\prec z)$, and for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then making use of the inductive definition of $(\prec)$ (meaning $(\prec_{\mathcal{F}})$) and the induction hypothesis, $$(L(x) \prec L(z) \prec L(z)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(z|_n d) = L(y). )$$

QED## Linear order from linearization

The linearization map $(L)$ is used to define the linear ordering on the notes of the Zettelkasten. Denote the new ordering on $(\mathcal{F(\mathcal{D}^+)})$ by $(\lll)$. Then the linear ordering is given by

$$(

w\lll x \Leftrightarrow L(w) \prec L(x)

)$$ for $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

## The map $(L)$ extends to a monoid homomorphism $(\Sigma^*\rightarrow\Sigma^*)$

The map $(L)$ can be obtained from the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols.

$$(L(w) =

\begin{cases}

w& \text{if}\quad w\in\mathbb{Z}^+;\\

\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&

\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\

\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.

\end{cases} )$$ for $(w\in \Sigma)$. The string substitution is $(O(n))$ in the length $(|n|)$ of the ID.

## Counterexamples

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

Let $(\mathcal{G}=\mathcal{F}(\mathcal{E}), \mathcal{E}=\mathcal{D}_{\ne0})$.

Since the nonzero decimals are dense, branches of $(\mathcal{G})$ are dense.

$$(\begin{array}{}

\forall x\in\mathcal{G},n\in\mathbb{Z}^+,c,d\in\mathcal{E}\\

\quad\left(c\prec 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\end{array})$$

However, the extension of the map $(L)$ to $(\mathcal{G})$ is not injective. Similarly for the extension of the map $(L)$ to the normalized IDs.

GitHub. Erdős #2. CC BY-SA 4.0.

## 20211017124743 Before I repair to Niflheim

A few paragraphs of comments about Folgezttel Formalized in this note were obliterated by the accidental selection and replacement of all of the text, instead of a small portion. This is easy to do on an iPad. Before I could revert to the previous draft, auto save kicked in. I know I could edit offline and upload, but be grateful that I didn't.

The main point: illustrations (preferably programmed in LaTeX) and an interpretation of the notation must be included. There were other remarks about the last minute change of the interpretation of the Folgezttel

`21/2a5b19f`

in the general IDs, but now that will have to wait. The introduction could be edited, references added. The use of Markdown comment syntax in callouts isn't optimal: they scroll horizontally if the text length exceeds the column width. Not a terrific idea to assume that readers will know this. Use something else or get rid of them.Auto save is forcing practice in concision and is reminding me that the reader's time is valuable.

As Hakuin is reported to have said, "is that so?"

`21/2a5b19f`

the 6th comment in a sequence of successive continuations of notes, starting with`21/2a5b19a`

, which is a comment on an aspect of`21/2a5b19`

, but is it necessarily the case that every member of the sequence comments on an aspect of`21/2a5b19`

? I am reluctant to commit to this conclusion. Nothing hangs on it as far as the general IDs go. Of course it can be stipulated to hold for some ZK.I had forgotten about this Markdown quirk.

GitHub. Erdős #2. CC BY-SA 4.0.

Re: footnotes -- the forum is "dumb" in that regard. Your footnote itself becomes the link. So if two or more posts are reachable via the same link, i.e. footnote "1", the target furthest at the top wins. That's how anchor links in HTML work. It'd be more useful if the forum applied some postprocessing so that each footnote is unique, but that's not the case. It also happened on our blog some years ago -- you wouldn't notice the problem when you read 1 article, because there, footnotes are unique on the page, but in an overview of multiple posts, whoops

Author at Zettelkasten.de • https://christiantietze.de/

## 20211021195316 Folgezettel Formalized

F Lengyel

#folgezettel #niklasluhmann #combinatorics #poset

## Introduction

A formalization and generalization is given of the system of Zettel identifiers (IDs) developed by the sociologist Niklas Luhmann. Luhmann's system of IDs, sometimes called Folgezttel, enabled Luhmann to maintain a dynamically branching tree structure within the linear ordering of the notes of his Zettelkasten. Each note added to the collection is assigned an ID indicating a place in the collection where the note either continues a line of thought in a preceding note; comments on or raises a question about some aspect of a preceding note; or starts a new topic. Further details on Luhmann's Zettelkasten and his system of IDs are given in (Schmidt, J. 2016, 2018).

The partially ordered set of generalized Folgezettel IDs is defined and is shown to specialize to Luhmann's Folgezttel IDs, up to renaming. Generalized Folgezettel IDs are coordinates of the nodes of an outline-like tree structure that may branch into any number of other such trees at any node or descendant node. The generalized Folgezettel IDs have the form $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$, where the $(v_i)$ are decimals, i.e., $(j)$-tuples of natural numbers written in the form $(n_1\mathbf{.}n_2\mathbf{.}\ldots\mathbf{.} n_{j})$. Such decimals are familiar as the section and subsection numbers of numbered outlines.

An order-preserving bijection is defined from the partially-ordered set of positive Folgezettel IDs (generalized Folgezettel IDs in which all coordinates are positive), to the lexicographically ordered set of normalized decimals, which are decimals in which the initial and final numbers are positive. This linearization map is used to define a linearization of the partially ordered set of positive Folgezettel IDs. The linearization generalizes the internal branching property of Luhmann's Folgezettel. The linearization map lifts to a monoid homomorphism on the monoid of words generated by the symbols of the language of the generalized Folgezettel IDs. This yields an efficient substitution algorithm for linearizing the partial order.

## Formalization

## 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 (

akathe 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

decimalis an element of the set $(\mathcal{D}_0)$. A decimal isnonzeroif at least one of its integer symbols is nonzero; the set of nonzero decimals is denoted $(\mathcal{D}_{\ne0})$. A decimal isnormalizedif its first and last integer symbols are nonzero; the set of normalized decimals is denoted $(\mathcal{D})$. A decimal ispositiveif all of its integer symbols are nonzero; the set of positive decimals is denoted $(\mathcal{D}^+)$.## Examples

$(0, 0\mathbf{.}0, 0\mathbf{.}1\mathbf{.}0\mathbf{.}1, 4\mathbf{.}0\mathbf{.}0, 1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}_0)$

$(0\mathbf{.}1\mathbf{.}0\mathbf{.}, 4\mathbf{.}0\mathbf{.}0, 1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}_{\ne0})$

$(1,400,25\mathbf{.}0\mathbf{.}0\mathbf{.}0.1,1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D})$

$(1, 1\mathbf{.}1\mathbf{.}1\mathbf{.}1, 29\mathbf{.}396\mathbf{.}4\mathbf{.}8,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}^+)$

## Lexicographic order on decimals

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is defined for $(u,v\in \mathcal{D}_0)$ by $$(u \prec v \Leftrightarrow \begin{cases} \exists x,y,x\in\mathcal{D}_0\cup\left\lbrace\varepsilon\right\rbrace,m,n\in\mathbb{N}, \\\quad\left(u=x\mathbf{.}m\mathbf{.}y\right) \land \left(v=x\mathbf{.}n\mathbf{.}z\right) \land \left(m\lt n\right); \\ \exists x\in\mathcal{D}_0, v = u\mathbf{.}x. \end{cases})$$

Note that in the second alternative above, $(x\ne\varepsilon.)$ The order $(\prec)$ is a lexicographic order, as is the associated partial order $(\preceq)$ defined by $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Each of the structures $$(\left(\mathcal{D}_0,\preceq\right),

\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),

\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),

\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ is lexicographically ordered by a sub-ordering of $(\preceq)$ restricted to the respective domain. We denote any of the lexicographic orders by $(\preceq)$, or by $(\preceq_\mathcal{D}, \preceq_{\mathcal{D}^+})$ as needed.

## Examples

$$(0\prec 0\mathbf{.}0\prec 0\mathbf{.}0\mathbf{.}0\mathbf{.}0\mathbf{.}0 \prec0\mathbf{.}1\prec 0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\prec1\mathbf{.}0\mathbf{.}0\mathbf{.}1 \prec2\mathbf{.}0\prec 2\mathbf{.}0\mathbf{.}0)$$

The nonzero condition rules out zero and "infinitesimals," which are decimals (except for zero) in which every digit is zero.

Note: The linearization of the Folgezettel IDs in the sequel will rely on the positive and normalized decimals $(\mathcal{D}^+)$ and $(\mathcal{D},)$ respectively.A

generalized Folgezettel ID(or simply, a 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 $(v_1,\ldots, v_k\in\mathcal{D}^+)$ are positive. Folgezettel IDs with nonzero and normalized decimals will serve as counterexamples to the linearization given in the sequel.The set of Folgezettel IDs is denoted by $(\mathcal{F})$. Observe that for $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k\in\mathcal{F}, )$ $(w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m \in\mathcal{F})$, $$(\left(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k = w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m\right) \Leftrightarrow \left(k=m \land i_n =j_n\, \land v_n=w_n\right))$$ for $(1\le n\le m\text{.})$

## Inductive definition of Folgezettel IDs

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 set of Folgezettel IDs is then $(\mathcal{F}=\mathcal{F}(\mathcal{D}^+).)$ From now on elements of $(\mathcal{F})$ will be referred to as IDs.

## Luhmann IDs

Notation: write $(`/\text{'})$ for the word $(`\left.\right|_1\text{'})$. An ID $(w\in\mathcal{F})$ is

Luhmann-likeif $$(w= m\mathbf{.}n / d_1 /\cdots / d_k,)$$ where $(\exists m, n, k, \in\mathbb{Z}^+, d_j\in\mathbb{Z}^+, 1\le j\le k.)$ The Luhmann-like IDs admit a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.## Translation from Luhmann's Folgezettel to Luhmann-like IDs

The translation to and from Luhmann's Folgezettel to Luhmann-like IDs will be given by example. Consider the Folgezettel ID $(21/2a5b19f)$. The corresponding Luhmann-like ID is $$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6.)$$ This ID is obtained by replacing the slash "/" with a period, and from the following alternating sequence of letters and numbers by inserting a slash "/" between each contiguous sequence of numbers (letters), and by replacing a letter (or letter sequence, such as "aa", which follows "z") by its corresponding ordinal value in lexicographic order. Thus, $(21/2a5b19f)$ becomes the expression $(21\mathbf{.}2/a/5/b/19/f)$, which then becomes $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$.

The interpretation of $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$ (and hence of $(21/2a5b19f)$) is the 6-th note in a sequence of notes starting with the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$, which itself comments on (or raises a question about) the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19)$. In the notation developed here, the note with decimal ID $(21\mathbf{.}2)$ represents the second note of a sequence starting with $(21\mathbf{.}1)$, under a category (or section) numbered $(21)$.

The reverse translation of a Luhmann-like note ID, such as $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6)$ (or without the syntactic sugar, $(21 \mathbf{.}2 \left.\right|_1 1 \left.\right|_1 5 \left.\right|_1 2 \left.\right|_1 19 \left.\right|_1 6)$), is the reverse process, in which the two-place decimal is replaced with the first number, a slash and the second number. The following slashes are removed in order, where letter sequences replacing numbers alternate with the next adjacent number, and so on until the end. Diagramatically,

$$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6 \rightarrow 21 / 2 a /5 / 2 /19 / 6\rightarrow 21 / 2 a 5 b /19 / 6\rightarrow 21 / 2 a 5 b 19 f)$$

Other embeddings of Luhmann's Folgezettel within the generalized IDs are possible.

## Partial order on IDs

The IDs have the structure of a partially ordered set $(\left(\mathcal{F}, \preceq_\mathcal{F}\right))$, where the partial order $(\preceq_\mathcal{F})$ on $(\mathcal{F})$ extends the relation $(\preceq_{\mathcal{D}^+})$ on $(\mathcal{D}^+)$, as follows.

For $(x, y\in\mathcal{F},)$ define $(x\prec_{\mathcal{F}} y)$ by induction on $(y)$. $$(x\prec_{\mathcal{F}} y\Leftrightarrow \begin{cases} x\prec_{\mathcal{D}^+} y, & x,y\in\mathcal{D}^+;\\ y= x\left.\right|_n d, &\exists n\in\mathbb{Z}^+, d\in\mathcal{D}^+;\\x=z\left.\right|_n c, y= z\left.\right|_n d, &\exists z\in\mathcal{F}, n\in\mathbb{Z}^+,c, d\in\mathcal{D}^+,c\prec_{\mathcal{D}^+}d;\\ y= z\left.\right|_n d,&\exists z\in\mathcal{F}, x\prec_{\mathcal{F}}z, n\in\mathbb{Z}^+, d\in\mathcal{D}^+.\end{cases})$$

The four cases of the preceding definition correspond respectively to

1- the comparison of (positive) decimals;

2- the comparison of an ID with a child (i.e., an immediate descendant of the ID);

3- the comparison of IDs on the same branch (i.e., they share the same initial segment ID);

4- the comparison of an ID with a remote descendant.

An inductive argument shows that these cases imply that $(\prec_{\mathcal{F}})$ is transitive.

Fact. If $(x\in\mathcal{F})$, $(y\in\mathcal{D}^+)$ and $(x\prec_{\mathcal{F}^+} y)$, then $(x\in\mathcal{D}^+)$. The proof is immediate, since $(x\in\mathcal{D}^+)$ is the only applicable case.## Linearization

Proposition. There is a bijective, order-preserving map $$(L: \left(\mathcal{F}, \preceq_{\mathcal{F}}\right)\rightarrow \left(\mathcal{D},\preceq_{\mathcal{D}^+}\right))$$ from the partially-ordered set of IDs to the lexicographically ordered set of normalized decimals. The map $(L)$ is not an order isomorphism.Proof. The map $(L)$ is inductively defined by $$(L(w) =

\begin{cases}

w,& w\in\mathcal{D}^+;\\

L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d, &\exists x\in\mathcal{F}^+, n\in\mathbb{Z}^+,d\in\mathcal{D}^+,

w= x\left.\right|_n d.

\end{cases} )$$ for $(w\in \mathcal{F})$.

## $(L)$ is well defined.

The map $(L)$ is well-defined (meaning that $(L)$ takes values in the indicated codomain): if $(w\in\mathcal{D}^+)$ then $(L(w)\in\mathcal{D})$. Let $(w\in\mathcal{F})$ and make the induction hypothesis that $$(\forall x\in\mathcal{F}, x\prec w\Rightarrow L(x)\in\mathcal{D}.)$$ Now suppose that the second case of the definition of $(L)$ holds, with $(w= x\left.\right|_n d)$, where $(x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D}.)$ Then $(x\prec w)$ (another induction, by definition of $(\prec)$) and by the induction hypothesis, $(L(x))$ is a normalized decimal; in particular, its initial word is nonzero. Since $(d)$ is a positive (hence normalized) decimal, its final word is nonzero, and therefore the value $$(L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d)$$ of $(L(w))$ is also a normalized decimal.

## $(L)$ is injective

To show that $(L)$ is injective, there are three cases for $(v,w\in\mathcal{F})$.

Case 1. $(v,w\in\mathcal{D}^+.)$

Case 2. $(v\in\mathcal{D}^+, w = x\left.\right|_n d)$ for $(x\in\mathcal{F},n\in\mathbb{Z}^+, \in\mathcal{D}^+.)$

Case 3. $(v = y\left.\right|_m c, w = z\left.\right|_n d)$, for $(y,z\in\mathcal{F},m,n\in\mathbb{Z}^+, c,d\in\mathcal{D}^+.)$

Make the induction hypothesis that for $(v,w\in\mathcal{F})$, $(L(v) = L(w) \Rightarrow v = w)$ by induction on $(w)$.

In Case 1, $(L(v) = L(w)\Rightarrow v = w)$ is immediate, since $(L)$ is the identity on $(\mathcal{D}^+)$.

In Case 2, suppose that $(L(v) = L(w))$. By definition of $(L)$,

$$(v=L(x\left.\right|_n d) = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d,)$$ which is impossible since $(v\in\mathcal{D}^+)$ ($(v)$ is a positive decimal) and $(L(w)\in \mathcal{D}\setminus\mathcal{D}^+)$ ($(L(w))$ is a normalized non-positive decimal).

In Case 3, suppose that $(L(y\left.\right|_m c) = L(z\left.\right|_n d).)$ Then,

$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}} c =

L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d. )$$ Now we reason in $(\Sigma^*)$. Since $(c, d)$ are positive decimals, $(c = d)$, so they can be cancelled, leaving$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}}= L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}. )$$ Since $(L(y), L(z))$ are normalized decimals (by the induction above), their final words are nonzero, so that $(m=n.)$ It follows that $(L(y) = L(z))$ and by the induction hypothesis, $(y = z)$. This yields injectivity in Case 3: $$( v = y\left.\right|_m c = z\left.\right|_n d = w)$$.

## $(L)$ is surjective.

To show that $(L)$ is surjective, there are two cases, the first being $(c\in\mathcal{D}^+)$, which is immediate ($(L(c)=c)$). If $(c\in\mathcal{D}\setminus\mathcal{D}^+)$, then $(c)$ is normalized but not positive, there is a rightmost consecutive string $(s)$ of zeros of length $(n=|s|\in\mathbb{Z}^+)$ between the initial and final words of $(c)$. This means we can write $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d,)$$ where $(b)$ is a normalized decimal and where $(d)$ is a positive decimal. Make the induction hypothesis on normalized decimals $(c)$, which is that surjectivity holds whenever $(b\in\mathcal{D})$ with $(b\prec c)$, and in that case $(b = L(x))$ for some $(x\in\mathcal{F})$. Then by definition of the map $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x\left.\right|_n d),)$$ where $(x\left.\right|_n d\in\mathcal{F})$. So $(L)$ is surjective.

## $(L)$ is order-preserving.

Finally, to show that $(L)$ is order-preserving, let $(x,y\in\mathcal{F})$ with $(x\prec y)$. We proceed by induction on $(y)$. The induction hypothesis (with parameter $(x\in\mathcal{F})$) is that for $(z\in\mathcal{F})$ with $(z\prec y)$, $$(x\prec z \Rightarrow L(x)\prec L(z).)$$ There are three cases.

Case 1. The case of $(y\in\mathcal{D}^+, x\prec y)$ forces $(x\in \mathcal{D}^+)$, which yields $(L(x)=x\prec y=L(y))$.

Case 2. If $(y = x |_n d)$ for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then $(x\prec x|_n d)$ so that $$(L(x) \prec L(x) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x |_n d) = L(y). )$$

Case 3. If $(y = z |_n d)$ for some $(z\in\mathcal{F})$ with $(x\prec z)$, and for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then making use of the inductive definition of $(\prec)$ (meaning $(\prec_{\mathcal{F}})$) and the induction hypothesis, $$(L(x) \prec L(z) \prec L(z)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(z|_n d) = L(y). )$$

QED## Linear order from linearization

The linearization map $(L)$ is used to define the linear ordering on the notes of the Zettelkasten. Denote the new ordering on $(\mathcal{F(\mathcal{D}^+)})$ by $(\lll)$. Then the linear ordering is given by$$(w\lll x \Leftrightarrow L(w) \prec L(x) )$$ for $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

## The map $(L)$ extends to a monoid homomorphism $(\Sigma^*\rightarrow\Sigma^*)$

The map $(L)$ can be obtained from the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols. $$(L(w) =\begin{cases}

w& \text{if}\quad w\in\mathbb{Z}^+;\\

\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&

\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\

\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.

\end{cases} )$$ for $(w\in \Sigma)$. The string substitution is $(O(n))$ in the length $(|n|)$ of the ID.

## Counterexamples

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

Let $(\mathcal{G}=\mathcal{F}(\mathcal{E}), \mathcal{E}=\mathcal{D}_{\ne0})$.

Since the nonzero decimals are dense, branches of $(\mathcal{G})$ are dense.

$$(\begin{array}{}

\forall x\in\mathcal{G},n\in\mathbb{Z}^+,c,d\in\mathcal{E}\\

\quad\left(c\prec 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\end{array})$$

However, the extension of the map $(L)$ to $(\mathcal{G})$ is not injective. Similarly for the extension of the map $(L)$ to the normalized IDs.

Acknowledgments. Thanks to @ctietze, @Sascha and @Will for helpful comments, criticism and corrections.## References

Schmidt, J. (2016). Niklas Luhmann‘s Card Index: Thinking Tool, Communication Partner, Publication Machine. In A. Cevolini (Ed.), Library of the written word: Vol. 53. Forgetting Machines. Knowledge Management Evolution in Early Modern Europe (1. Auflage., pp. 289-311). Leiden: Brill.

Schmidt, J. F. (2018). Niklas Luhmann’s Card Index: The Fabrication of Serendipity. Sociologica, 12(1), 53–60. https://doi.org/10.6092/issn.1971-8853/8350

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

GitHub. Erdős #2. CC BY-SA 4.0.

## 20211021210810 Another Draft, Another Dollar

After reading "Writing Philosophy" by Richard A. Watson, which was recommended in comments to a post in the Leiter Reports blog, I wrote another draft of Folgezettel Formalized While this isn't philosophy and I am not a philosopher; the math is trivial enough to meet professional standards for mathematical depth within philosophy; Luhmann's theoretical sociology tends to be viewed as philosophy within the Anglophone community of scholars (not to rain on anyone's parade—there are rebuttals in the literature to the claim that Luhmann's work is not even wrong, but none that I am aware of venture to bring Luhmann's abstractions into contact with the Earth—though someone might succeed, God willing); and I thought Prof Watson book was applicable, and pretty well written.

I did take issue with Watson's recommendation to consult the linguistically incompetent

Elements of Styleby Strunk and White, when the competent and vastly superiorStyle: Lessons in Clarity and Graceby Joseph M. Williams is available. Also, I don't insist on avoiding the singular 'they', and I am prepared to haul out the 1000+ page descriptivistCambridge Grammar of the English Languageby Rodney Huddleston and Geoffrey Pullum in the almost-lethal grammatical struggle for survival against all fustian prescriptivist adversaries. Unless it means not getting published. Then I will appear to be a completely different person. Following Richard A. Watson's advice to avoid jokes, I decided that the post could convulse its readers into paroxysms of inextinguishable laughter by itself, without my efforts at humor.Now I am abandoning the LaTeX package xyPic for diagrams in TikZ. This is easy, since xyPic wasn't memorable. Ghosted, after a couple of decades of loyal LaTeX package inclusion, just like that. I will attempt to develop TikZ diagrams for the next draft.

GitHub. Erdős #2. CC BY-SA 4.0.

## 20211023135158 A better introductory sentence

This is exactly what Folgezettel IDs are mathematically: coordinates of a skeletal backbone for a Zettelkasten. The description occurred to me while I was rewriting a Zettel on a Zettel format within Zettlr that would use Folgezettel, in addition to timestamp IDs. (At this level and this rate, I have long since perished.)

A possible revision. In general, there is a forest of spanning trees within a Zettelkasten. We should be careful not to assume that the graph of Zettels is connected.

Add references to this site and to the book by @Sascha. @Sascha is the preceding paragraph correct? At some point we should ask Johannes Schmidt, one of the world's experts on the Luhmann archive, whether this is correct.

GitHub. Erdős #2. CC BY-SA 4.0.

## 12021102321344 A mathcha.io generated TikZ diagram

Here is a diagram to illustrate that general Folgezettel are coordinates of indefinitely nested and parallel outlines. The diagram was generated using the https://mathcha.io online interactive editor, hosted in the Republic of Singapore.

The figure below contains a "top level" outline with sections numbered $(1)$, $(1.1)$ and $(1.2)$. The node $(1.1)$ has two comments, indicated by $(1.1|_1 1)$, indicating the first comment on an aspect of $(1.1)$, and $(1.1|_3 1)$, indicating the third comment on aspect of $(1.1)$. The Zettel with ID $(1.1|_1 1.1 |_1 1)$ is the first comment on an aspect of $(1.1|_1 1.1)$. Likewise, $(1.1|_3 1|_5 1)$ is the fifth comment on $(1.1|_3 1)$, and

$(1.1|_3 1|_5 2.5.3)$ is node $(2.5.3)$ within an outline that begins with $(1.1|_3 1|_5 1)$. To illustrate that the numbering handles forests of trees, there is the purple tree.

## The terms of the Folgezettel debate

Not shown are internal links within Zettels. However, the formalization illustrates and clarifies what the Folgezettel ID debate is about. A Zettelkasten developed with Folgezettel IDs has a distinguished spanning tree for each of its connected components (in the absence of other links connecting those trees). The term "distinguished" indicates that a specific spanning tree is named by the Folgezettel IDs— these are present in the diagram. In these terms, the Folgezettel debate concerns whether a distingushed spanning tree "backbone" for a Zettelkasten offers the researcher or writer a significant advantage over a Zettelkasten lacking a distinguished spanning tree.

I'm not taking a position. There may be no benefit (that's the null hypothesis); there might be some benefit; there might be an overwhelming payoff with an immense measurable effect size; the payoff could be worse than no Zettelkasten; one might find that structure notes always beat Folgezettel; or perhaps most people would be better off with an "invertebrate" Zettelkasten. In the absence of a framework to make such judgments, I can't say myself.

## Finding spanning trees subject to constraints

That there exist spanning trees in digital Zettelkasten with timestamp IDs (for example) is obvious. What isn't obvious is the computational complexity of finding and labelling spanning trees subject to constraints, such as whether there exists a spanning tree with a note labelling such that each note label reflects the decision that was taken when the note was added to the Zettelkasten.

## The general Folgezettel really are general

Restricting to one change of branch, i.e., $(z |_1 d)$ up to renumbering, or in words, only one comment per node, is topologically identical with Luhmann's system. In Luhmann's system, the degree of the nodes of the (undirected) spanning trees defined by Folgezettel is always at most $(3)$ (forgetting internal links--there's a forgetful functor for you, though mathematicians sometimes forget to define what a "forgetful functor" is).

This particular diagram cannot be represented with Luhmann's notation, since two comments on the Zettel with ID $(1.1)$ are shown, namely $(1.1|_1 1)$ and $(1.1|_3 1)$. Numerical invariants are more efficient: the degree (total in- plus out-degree if you wish) of $(1.1)$ and $(1.1|_3 1)$ is $(4)$, which of course is greater than the maximum degree $(3)$ of any ID representable with Luhmann's Folgezettel (again forgetting internal links).

GitHub. Erdős #2. CC BY-SA 4.0.

## 20211024164953 Effective Information in Zettelkasten Networks and the Folgezettel Debate

My Zettel title is somewhat pretentious: I haven't measured diddly in Zettelkasten networks. But I want to imagine a possible future.

Only yesterday I said that I knew of no scientific framework for addressing (much less resolving) the Folgezettel debate. Now I think this question isn't as narrow and specialized as it sounds.

One way of stating the question: does the presence of a distinguished spanning tree (the one named by Folgezettel) offer some measurable benefit to the researcher or writer? How do Zettelkasten organized around structure notes compare?

This is still vague, since the notion of "measurable benefit" hasn't been defined. I haven't said anything about the growth of such networks over time; I haven't attempted to define a measure of semantic content within individual Zettels (nevertheless I have some idea where to look); and I haven't stated a criterion for "organization around structure notes" that could even be workshopped, let alone survive peer review.

But applications of information theory to networks is an active area of research. If I were going to think more seriously about this, today I would start here:

Brennan Klein, Erik Hoel.

The emergence of informative higher scales in complex networks. arXiv:1907.03902[physics.soc-ph]I'll quote the abstract:

GitHub. Erdős #2. CC BY-SA 4.0.

Fellow grammarians unite! One of the threads that keeps me interested in this conversation is your brain-bewitching and sometimes dumbfounding humor. Just as the discussion gets a bit too technical and I get lost in the equations, you drop a pun or joke like dismounting my backpack after the climb into Amy Lake. A combination of the feeling of relief at the new sense of weightlessness and exhilaration because of the natural surroundings. Would you please continue with the jokes! Maybe the jokes can help flush ideas from your thesis. Watson's not watching.

Zettel is like moose, both singular and plural.

https://forum.zettelkasten.de/discussion/1455/q-a-6-plural-of-zettel

The "forgetful functor" reminds me of the proverbial curse of knowledge or:

Will Simpson

My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time.

kestrelcreek.com

This research looks like it might take us somewhere towards the notion of a "measurable benefit." The goal is to measure and incrementally improve the amount and usefulness of information carried in the connection between zettel. Foglezettel IDs is one way. Another is a structure note title/subtitle scheme called structure note.

Information contained in the connections includes, subsumes, and goes beyond the physicalness of the information in the outline syntax. How? I'm not sure? I'll have to look at this paper closer. Especially section IV-A. Selection of real networks. The graphs, created in python, are eye-candy for a budding data spelunker.

A. Koseska and P. I. H. Bastiaens, Cell signaling as a cognitive process, The EMBO Journal 36, 568 (2017). ↩︎

F. A. Rodrigues, T. K. D. M. Peron, P. Ji, and J. Kurths, The Kuramoto model in complex networks, Physics Reports 610, 1 (2016). ↩︎

Will Simpson

My peak cognition is behind me. One day I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time.

kestrelcreek.com

Well, yes. That's why I mentioned it. Cost benefit would be even better.

I would start by computing this measure on existing ZK networks, to see whether it yields anything useful. Luhmann's ZK would be a good test case, again assuming the measure will yield something useful. If the measure returns the same value one as a one node network as it does with Luhmann's ZK (unlikely), then either the measure isn't useful or else the measure doesn't detect what made Luhmann's ZK network useful to Luhmann.

At least before attempting to use it for incremental improvements, I would attempt to get some feeling for the measure with examples.

That's my methodological two cents, having computed the Kullback-Liebler divergence of probability distributions of climate-related data. Ideally one would avoid having a distinguished senior scientist look at your calculations and exclaim, "Where are the units in these graphs? This has no scientific interest." (I've seen this, let's say.)

GitHub. Erdős #2. CC BY-SA 4.0.

## 20211030133016 Folgezettel Formalized

F Lengyel

#folgezettel #niklasluhmann #combinatorics #poset

## Introduction

A formalization and generalization is given of the system of Zettel identifiers, sometimes called Folgezettel, developed by the sociologist Niklas Luhmann. Folgezettel IDs are spanning tree coordinates for the graph of Zettels of a Zettelkasten. The graph of Zettels may have several disconnected components, each with its own distingushed spanning tree. In addition to specifying the location of a Zettel on a spanning tree, a Folgezettel ID indicates whether that Zettel either: continues a prior Zettel; comments on or raises a question about an aspect of a prior Zettel; or begins a new topic. Niklas Luhmann's Zettelkasten and his system of Zettel IDs are described in (Schmidt, J. 2016, 2018).

The partially ordered set of generalized Folgezettel IDs is defined and is shown to specialize to Luhmann's Folgezttel IDs, up to renaming. Generalized Folgezettel IDs are coordinates of the nodes of an outline-like tree structure that may branch into any number of other such trees at any node or descendant node. The generalized Folgezettel IDs have the form $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$, where the $(v_i)$ are decimals, i.e., $(j)$-tuples of natural numbers written in the form $(n_1\mathbf{.}n_2\mathbf{.}\ldots\mathbf{.} n_{j})$. Such decimals are familiar as the section and subsection numbers of numbered outlines.

An order-preserving bijection is defined from the partially-ordered set of positive Folgezettel IDs (generalized Folgezettel IDs in which all coordinates are positive), to the lexicographically ordered set of normalized decimals, which are decimals in which the initial and final numbers are positive. This linearization map is used to define a linearization of the partially ordered set of positive Folgezettel IDs. The linearization generalizes the internal branching property of Luhmann's Folgezettel. The linearization map lifts to a monoid homomorphism on the monoid of words generated by the symbols of the language of the generalized Folgezettel IDs. This yields an algorithm for linearizing the partial order.

## Formalization

## 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 (

akathe 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

decimalis an element of the set $(\mathcal{D}_0)$. A decimal isnonzeroif at least one of its integer symbols is nonzero; the set of nonzero decimals is denoted $(\mathcal{D}_{\ne0})$. A decimal isnormalizedif its first and last integer symbols are nonzero; the set of normalized decimals is denoted $(\mathcal{D})$. A decimal ispositiveif all of its integer symbols are nonzero; the set of positive decimals is denoted $(\mathcal{D}^+)$.## Examples

$(0, 0\mathbf{.}0, 0\mathbf{.}1\mathbf{.}0\mathbf{.}1, 4\mathbf{.}0\mathbf{.}0, 1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}_0)$

$(0\mathbf{.}1\mathbf{.}0\mathbf{.}, 4\mathbf{.}0\mathbf{.}0, 1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}_{\ne0})$

$(1,400,25\mathbf{.}0\mathbf{.}0\mathbf{.}0.1,1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D})$

$(1, 1\mathbf{.}1\mathbf{.}1\mathbf{.}1, 29\mathbf{.}396\mathbf{.}4\mathbf{.}8,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}^+)$

## Lexicographic order on decimals

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is defined for $(u,v\in \mathcal{D}_0)$ by $$(u \prec v \Leftrightarrow \begin{cases} \exists x,y,x\in\mathcal{D}_0\cup\left\lbrace\varepsilon\right\rbrace,m,n\in\mathbb{N}, \\\quad\left(u=x\mathbf{.}m\mathbf{.}y\right) \land \left(v=x\mathbf{.}n\mathbf{.}z\right) \land \left(m\lt n\right); \\ \exists x\in\mathcal{D}_0, v = u\mathbf{.}x. \end{cases})$$

Note that in the second alternative above, $(x\ne\varepsilon.)$ The order $(\prec)$ is a lexicographic order, as is the associated partial order $(\preceq)$ defined by $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Each of the structures $$(\left(\mathcal{D}_0,\preceq\right),

\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),

\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),

\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ is lexicographically ordered by a sub-ordering of $(\preceq)$ restricted to the respective domain. We denote any of the lexicographic orders by $(\preceq)$, or by $(\preceq_\mathcal{D}, \preceq_{\mathcal{D}^+})$ as needed.

## Examples

$$(0\prec 0\mathbf{.}0\prec 0\mathbf{.}0\mathbf{.}0\mathbf{.}0\mathbf{.}0 \prec0\mathbf{.}1\prec 0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\prec1\mathbf{.}0\mathbf{.}0\mathbf{.}1 \prec2\mathbf{.}0\prec 2\mathbf{.}0\mathbf{.}0)$$

The nonzero condition rules out zero and "infinitesimals," which are decimals (except for zero) in which every digit is zero.

Note: The linearization of the Folgezettel IDs in the sequel will rely on the positive and normalized decimals $(\mathcal{D}^+)$ and $(\mathcal{D},)$ respectively.A

generalized Folgezettel ID(or simply, a Folgezettel ID) is a word of $(\Sigma^*)$ of the orm $$(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$$ where $(k\in\mathbb{Z}^+,i_1,\ldots, i_k\in\mathbb{Z}^+)$ and where the decimals $(v_1,\ldots, v_k\in\mathcal{D}^+)$ are positive. Folgezettel IDs with nonzero and normalized decimals will serve as counterexamples to the linearization given in the sequel.The set of Folgezettel IDs is denoted by $(\mathcal{F})$. Th following diagram illustrates that Folgezettel IDs are coordinates of the nodes of a forest of outline-like tree structures that may branch into any number of other such trees at any node or descendant node. Internal links between nodes are not shown in this diagram.

The notation shown will be interpreted in terms of the choices made when a new node is added top the forest of trees. The figure above contains a "top level" outline with sections numbered $(1)$, $(1.1)$ and $(1.2)$. The node $(1.1)$ has two comments, indicated by $(1.1|_1 1)$, indicating the first comment on an aspect of $(1.1)$, and $(1.1|_3 1)$, indicating the third comment on aspect of $(1.1)$. The Zettel with ID $(1.1|_1 1.1 |_1 1)$ is the first comment on an aspect of $(1.1|_1 1.1)$. Likewise, $(1.1|_3 1|_5 1)$ is the fifth comment on $(1.1|_3 1)$, and $(1.1|_3 1|_5 2.5.3)$ is node $(2.5.3)$ within an outline that begins with $(1.1|_3 1|_5 1)$. The purple tree illustrates that the numbering handles forests of trees.

## Inductive definition of Folgezettel IDs

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 set of Folgezettel IDs is then $(\mathcal{F}=\mathcal{F}(\mathcal{D}^+).)$ From now on elements of $(\mathcal{F})$ will be referred to as IDs.

## Luhmann IDs

Notation: write $(`/\text{'})$ for the word $(`\left.\right|_1\text{'})$. An ID $(w\in\mathcal{F})$ is

Luhmann-likeif $$(w= m\mathbf{.}n / d_1 /\cdots / d_k,)$$ where $(\exists m, n, k, \in\mathbb{Z}^+, d_j\in\mathbb{Z}^+, 1\le j\le k.)$ The Luhmann-like IDs admit a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.## Translation from Luhmann's Folgezettel to Luhmann-like IDs

The translation to and from Luhmann's Folgezettel to Luhmann-like IDs will be given by example. Consider the Folgezettel ID $(21/2a5b19f)$. The corresponding Luhmann-like ID is $$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6.)$$ This ID is obtained by replacing the slash "/" with a period, and from the following alternating sequence of letters and numbers by inserting a slash "/" between each contiguous sequence of numbers (letters), and by replacing a letter (or letter sequence, such as "aa", which follows "z") by its corresponding ordinal value in lexicographic order. Thus, $(21/2a5b19f)$ becomes the expression $(21\mathbf{.}2/a/5/b/19/f)$, which then becomes $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$.

The interpretation of $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$ (and hence of $(21/2a5b19f)$) is the 6-th note in a sequence of notes starting with the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /1)$, which itself comments on (or raises a question about) the note with ID $(21 \mathbf{.} 2 / 1 /5 / 2 /19)$. In the notation developed here, the note with decimal ID $(21\mathbf{.}2)$ represents the second note of a sequence starting with $(21\mathbf{.}1)$, under a category (or section) numbered $(21)$.

The reverse translation of a Luhmann-like note ID, such as $(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6)$ (or without the syntactic sugar, $(21 \mathbf{.}2 \left.\right|_1 1 \left.\right|_1 5 \left.\right|_1 2 \left.\right|_1 19 \left.\right|_1 6)$), is the reverse process, in which the two-place decimal is replaced with the first number, a slash and the second number. The following slashes are removed in order, where letter sequences replacing numbers alternate with the next adjacent number, and so on until the end. Diagramatically,

$$(21 \mathbf{.} 2 / 1 /5 / 2 /19 /6 \rightarrow 21 / 2 a /5 / 2 /19 / 6\rightarrow 21 / 2 a 5 b /19 / 6\rightarrow 21 / 2 a 5 b 19 f)$$

## The general Folgezettel IDs versus Luhmann's Folgezettel

Restricting to one change of branch, i.e., $(z |_1 d)$ up to renumbering, or in words, only one comment per node, is topologically identical with Luhmann's system. In Luhmann's system, the degree of the nodes of the (undirected) spanning trees defined by Folgezettel is always at most $(3)$. A simple degree argument shows that diagram above cannot be represented with Luhmann's notation. The degree (total in- plus out-degree if you wish) of $(1.1)$ and $(1.1|_3 1)$ is $(4)$, which of course is greater than the maximum degree $(3)$ of any ID representable with Luhmann's Folgezettel (again forgetting internal links).

Other embeddings of Luhmann's Folgezettel within the generalized IDs are possible.

## Partial order on IDs

The IDs have the structure of a partially ordered set $(\left(\mathcal{F}, \preceq_\mathcal{F}\right))$, where the partial order $(\preceq_\mathcal{F})$ on $(\mathcal{F})$ extends the relation $(\preceq_{\mathcal{D}^+})$ on $(\mathcal{D}^+)$, as follows.

For $(x, y\in\mathcal{F},)$ define $(x\prec_{\mathcal{F}} y)$ by induction on $(y)$. $$(x\prec_{\mathcal{F}} y\Leftrightarrow \begin{cases} x\prec_{\mathcal{D}^+} y, & x,y\in\mathcal{D}^+;\\ y= x\left.\right|_n d, &\exists n\in\mathbb{Z}^+, d\in\mathcal{D}^+;\\x=z\left.\right|_n c, y= z\left.\right|_n d, &\exists z\in\mathcal{F}, n\in\mathbb{Z}^+,c, d\in\mathcal{D}^+,c\prec_{\mathcal{D}^+}d;\\ y= z\left.\right|_n d,&\exists z\in\mathcal{F}, x\prec_{\mathcal{F}}z, n\in\mathbb{Z}^+, d\in\mathcal{D}^+.\end{cases})$$

The four cases of the preceding definition correspond respectively to

1. the comparison of (positive) decimals;

2. the comparison of an ID with a child (i.e., an immediate descendant of the ID);

3. the comparison of IDs on the same branch (i.e., they share the same initial segment ID);

4. the comparison of an ID with a remote descendant.

An inductive argument shows that these cases imply that $(\prec_{\mathcal{F}})$ is transitive.

Fact. If $(x\in\mathcal{F})$, $(y\in\mathcal{D}^+)$ and $(x\prec_{\mathcal{F}^+} y)$, then $(x\in\mathcal{D}^+)$. The proof is immediate, since $(x\in\mathcal{D}^+)$ is the only applicable case.## Linearization

Proposition. There is a bijective, order-preserving map $$(L: \left(\mathcal{F}, \preceq_{\mathcal{F}}\right)\rightarrow \left(\mathcal{D},\preceq_{\mathcal{D}^+}\right))$$ from the partially-ordered set of IDs to the lexicographically ordered set of normalized decimals. The map $(L)$ is not an order isomorphism.Proof. The map $(L)$ is inductively defined by $$(L(w) =\begin{cases}

w,& w\in\mathcal{D}^+;\\

L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d, &\exists x\in\mathcal{F}^+, n\in\mathbb{Z}^+,d\in\mathcal{D}^+,

w= x\left.\right|_n d.

\end{cases} )$$ for $(w\in \mathcal{F})$.

## $(L)$ is well defined.

The map $(L)$ is well-defined (meaning that $(L)$ takes values in the indicated codomain): if $(w\in\mathcal{D}^+)$ then $(L(w)\in\mathcal{D})$. Let $(w\in\mathcal{F})$ and make the induction hypothesis that $$(\forall x\in\mathcal{F}, x\prec w\Rightarrow L(x)\in\mathcal{D}.)$$ Now suppose that the second case of the definition of $(L)$ holds, with $(w= x\left.\right|_n d)$, where $(x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D}.)$ Then $(x\prec w)$ (another induction, by definition of $(\prec)$) and by the induction hypothesis, $(L(x))$ is a normalized decimal; in particular, its initial word is nonzero. Since $(d)$ is a positive (hence normalized) decimal, its final word is nonzero, and therefore the value $$(L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d)$$ of $(L(w))$ is also a normalized decimal.

## $(L)$ is injective

To show that $(L)$ is injective, there are three cases for $(v,w\in\mathcal{F})$.

Case 1. $(v,w\in\mathcal{D}^+.)$

Case 2. $(v\in\mathcal{D}^+, w = x\left.\right|_n d)$ for $(x\in\mathcal{F},n\in\mathbb{Z}^+, \in\mathcal{D}^+.)$

Case 3. $(v = y\left.\right|_m c, w = z\left.\right|_n d)$, for $(y,z\in\mathcal{F},m,n\in\mathbb{Z}^+, c,d\in\mathcal{D}^+.)$

Make the induction hypothesis that for $(v,w\in\mathcal{F})$, $(L(v) = L(w) \Rightarrow v = w)$ by induction on $(w)$.

In Case 1, $(L(v) = L(w)\Rightarrow v = w)$ is immediate, since $(L)$ is the identity on $(\mathcal{D}^+)$.

In Case 2, suppose that $(L(v) = L(w))$. By definition of $(L)$,

$$(v=L(x\left.\right|_n d) = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d,)$$ which is impossible since $(v\in\mathcal{D}^+)$ ($(v)$ is a positive decimal) and $(L(w)\in \mathcal{D}\setminus\mathcal{D}^+)$ ($(L(w))$ is a normalized non-positive decimal).

In Case 3, suppose that $(L(y\left.\right|_m c) = L(z\left.\right|_n d).)$ Then,

$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}} c =

L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d. )$$ Now we reason in $(\Sigma^*)$. Since $(c, d)$ are positive decimals, $(c = d)$, so they can be cancelled, leaving$$( L(y) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{m\, \text{zeros}}= L(z) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}. )$$ Since $(L(y), L(z))$ are normalized decimals (by the induction above), their final words are nonzero, so that $(m=n.)$ It follows that $(L(y) = L(z))$ and by the induction hypothesis, $(y = z)$. This yields injectivity in Case 3: $$( v = y\left.\right|_m c = z\left.\right|_n d = w)$$.

## $(L)$ is surjective.

To show that $(L)$ is surjective, there are two cases, the first being $(c\in\mathcal{D}^+)$, which is immediate ($(L(c)=c)$). If $(c\in\mathcal{D}\setminus\mathcal{D}^+)$, then $(c)$ is normalized but not positive, there is a rightmost consecutive string $(s)$ of zeros of length $(n=|s|\in\mathbb{Z}^+)$ between the initial and final words of $(c)$. This means we can write $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d,)$$ where $(b)$ is a normalized decimal and where $(d)$ is a positive decimal. Make the induction hypothesis on normalized decimals $(c)$, which is that surjectivity holds whenever $(b\in\mathcal{D})$ with $(b\prec c)$, and in that case $(b = L(x))$ for some $(x\in\mathcal{F})$. Then by definition of the map $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x\left.\right|_n d),)$$ where $(x\left.\right|_n d\in\mathcal{F})$. So $(L)$ is surjective.

## $(L)$ is order-preserving.

Finally, to show that $(L)$ is order-preserving, let $(x,y\in\mathcal{F})$ with $(x\prec y)$. We proceed by induction on $(y)$. The induction hypothesis (with parameter $(x\in\mathcal{F})$) is that for $(z\in\mathcal{F})$ with $(z\prec y)$, $$(x\prec z \Rightarrow L(x)\prec L(z).)$$ There are three cases.

Case 1. The case of $(y\in\mathcal{D}^+, x\prec y)$ forces $(x\in \mathcal{D}^+)$, which yields $(L(x)=x\prec y=L(y))$.

Case 2. If $(y = x |_n d)$ for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then $(x\prec x|_n d)$ so that $$(L(x) \prec L(x) \underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(x |_n d) = L(y). )$$

Case 3. If $(y = z |_n d)$ for some $(z\in\mathcal{F})$ with $(x\prec z)$, and for some $(n\in\mathbb{Z}^+, d\in \mathcal{D}^+)$, then making use of the inductive definition of $(\prec)$ (meaning $(\prec_{\mathcal{F}})$) and the induction hypothesis, $$(L(x) \prec L(z) \prec L(z)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}} d = L(z|_n d) = L(y). )$$

QED## Linear order from linearization

The linearization map $(L)$ is used to define the linear ordering on the notes of the Zettelkasten. Denote the new ordering on $(\mathcal{F(\mathcal{D}^+)})$ by $(\lll)$. Then the linear ordering is given by$$(w\lll x \Leftrightarrow L(w) \prec L(x) )$$ for $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

## The map $(L)$ extends to a monoid homomorphism

The map $(L)$ can be obtained from the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols. $$(L(w) =\begin{cases}

w& \text{if}\quad w\in\mathbb{Z}^+;\\

\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&

\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\

\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.

\end{cases} )$$ for $(w\in \Sigma)$.

## Counterexamples

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

Let $(\mathcal{G}=\mathcal{F}(\mathcal{E}), \mathcal{E}=\mathcal{D}_{\ne0})$.

Since the nonzero decimals are dense, branches of $(\mathcal{G})$ are dense.

$$(\begin{array}{}

\forall x\in\mathcal{G},n\in\mathbb{Z}^+,c,d\in\mathcal{E}\\

\quad\left(c\prec 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\end{array})$$

However, the extension of the map $(L)$ to $(\mathcal{G})$ is not injective. Similarly for the extension of the map $(L)$ to the normalized IDs.

## Remarks on spanning trees and the Folgezettel debate

A Zettelkasten developed with Folgezettel IDs has a distinguished spanning tree for each of its connected components (in the absence of other links connecting those trees). The term "distinguished" indicates that a specific spanning tree is named by the Folgezettel IDs—such spanning trees are present in the above diagram, one for each the two trees of the forest shown. In these terms, the Folgezettel debate concerns whether a distinguished spanning tree "backbone" for a Zettelkasten offers the researcher or writer a significant advantage over a Zettelkasten lacking a distinguished spanning tree.

That there exist spanning trees in digital Zettelkasten with timestamp IDs (for example) is obvious. What isn't obvious is the computational complexity of finding and labelling spanning trees subject to constraints, such as finding a spanning tree with a note labelling such that each note label reflects the decision that was taken when the note was added to the Zettelkasten.

Acknowledgments. Thanks to @ctietze, @Sascha and @Will for helpful comments, criticism and corrections.## References

Schmidt, J. (2016). Niklas Luhmann‘s Card Index: Thinking Tool, Communication Partner, Publication Machine. In A. Cevolini (Ed.), Library of the written word: Vol. 53. Forgetting Machines. Knowledge Management Evolution in Early Modern Europe (1. Auflage., pp. 289-311). Leiden: Brill.

Schmidt, J. F. (2018). Niklas Luhmann’s Card Index: The Fabrication of Serendipity. Sociologica, 12(1), 53–60. https://doi.org/10.6092/issn.1971-8853/8350

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

GitHub. Erdős #2. CC BY-SA 4.0.

I think the Vienna Circle could help you.

But I understand too little math to be of help.

I think the issue is that the relationship between notes is determined by the relationship of the knowledge that you connect. The connection is not characterised by the fact that there is a connection or the figuration of links between the notes but because of the specific characteristics if you link to an illustrating anecdote or body of evidence, single study etc.

It is not structure notes in isolation. It is structure note + the habit of starting note create with creating the link to the new note first.

I think it is an empirical question. I could argue (I don't) that Folgezettel lead to a false sense of accomplishment since you have one "link" garantueed. However, to embedd a new piece of knowledge or transform information into knowledge to be able to embedd is dependent on a bit more.

I wouldn't count so much on the element of suprise. It comes from the ability to connect thoughts over infinite time (theoretically) and not from within a moment.

I am a Zettler

My time machine is in the garage. But if the opportunity to move to Vienna ever presents itself, I would not look back.

I see this.

True also for structure notes.

I hadn't actually written this. There are measures at least between words.

This hasn't exactly aged well. Mostly superseded by 20211030133016 Folgezettel Formalized. I could add a my Zettel format to the thread.

GitHub. Erdős #2. CC BY-SA 4.0.

Earlier I mentioned that I would include my Zettel format, obtained after the usual futzing, second guessing, backtracking and compulsive oversharing common among ZK-maniacs. The Folgezettel IDs of Luhmann or of 20211030133016 above add too much friction in a digital ZK to be worth the effort. An adaptation of Folgezettel to digital ZK is proposed below. Unresolved links below refer to my own ZK. They aren't necessary to understand this post. I tend to postpone the use of structure notes to the bitter end. The philosophy isthink globally, link locally. Next: a checklist, in the spirit of The Checklist Manifesto.## 20210424152745 Zettel format: rule of threes

#zettel #zettelkasten #ruleofthrees #folgezettel #keywords #zettelformat

RELATED [[20210424174054]] Zettlr title format #+ImmutableID+Title, [[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

Header: in three plus 1 parts.`immutableID`

, followed by a title, referred to in this Zettel by`title`

;the keyword RELATED followed by a list of IDs of prior Zettels related to

`immutableID`

; see 1.c.2 below.1.a. YAML frontmatter: optional. If there are Pandoc-style references in the Zettel body, add the following to the YAML frontmatter header.

---

reference-section-title: References

---

1.b. Level 1 header including an immutableID and title.

SYNTAX

#

`immutableID`

`title`

1.c.1. Keywords in #hashtag format.

SYNTAX

#keyword #example

1.c.2. RELATED

SYNTAX

The keyword 'RELATED' followed by a comma-separated list of Zettel IDs such that for each

`ID`

in the list,`immutableID`

either:`ID`

;`ID`

; or,`ID`

.Body: an atomic note.`immutableID`

is added to RELATED.Footer: References.## References

Theory of Graded Consequence: a General Framework for Logics of Uncertainty. Springer Singapore, 2019.GitHub. Erdős #2. CC BY-SA 4.0.

You pulled off a thorough discussion of your example by displaying an example. Not the easiest thing to do, but as you told us what you were doing, you did precisely that. Anyone commenting on your zettel format would only be picking around the edges. I'm referring to that "futzing, second-guessing, backtracking, and compulsive oversharing ... ZK-maniac" @Will.

Yea, but ... Google tells me that "units of thought" are symbols, concepts, prototypes, images, muscular responses, and language. These originate in that there squishy grey matter bucket and are not relevant to NIST.

When I've done simple Keyboard Maestro scripting, I found that having a UUID with a different format from links and placing it, along with other stuff in the YAML front matter of each note makes scripting easy. This little trick creates a link that surfaces all the inbound links because each zettel has one and only one. It is an excellent hook for scripting.

This is a bit unsettled in my thinking. Is

`#zettelkasting`

an object or a category? Is`#thinking-skills`

an object or a category? They are more of an activity. I see the value tags add as something that is on a spectrum. Complex objects like the #universe, pointing at one end of the spectrum, is worthless for a tag. In comparison, a discrete category such as #objectophilia is specific enough to be of value in tying together the few possible zettel covering this category. In mathematics, categories and objects go together, but who am I to tell you this?kestrelcreek.com

I owe a lot to this community. The least I could do is complete what I set out to do.

Thank you. Most of my format survived, but thanks to your remarks, I'm going to revise.

After some reflection, my candidate for a "unit of thought" is the

schemaof Cognitive Load Theory (John Sweller). Thank you for prompting me to recall Cognitive Load Theory (CLT).As for NIST, perhaps as neural computer architecture advances, and provided Sweller's theory accurately predicts that schemata can be identified within the brain (somehow), in the remote future, the NIST museum will exhibit a brain-like neural architecture computer whose internal state demonstrably "encodes" the schemata of John Sweller's theory.

As @MartinBB reminds us, we don't know how the brain works. Nevertheless, bracketing a stupefying level of ignorance, in the nomenclature of CLT, the ideal ZK would help to:

How to put this into practice eludes me at the moment. The Cognitive Load-ites provide examples for educators of the three types of cognitive load (if I keep typing I'll manufacture my own load). For now I'll assume that CLT is true, and remark in the section on the Zettel body of the revision that the Zettel format and content should facilitate 1, 2 and 3 (because I assume this is feasible, worthwhile and better than the alternatives).

UPDATE: the voice of @Sascha in my head reminds me that the body of the Zettel should be written for the future self. It should be framed to get the future self to do what it is supposed to do. That's an additional constraint, or guideline. If we only knew how.

Add references to CLT, @Sascha and Frame It Again: New Tools for Rational Decision-Making by José Luis Bermúdez.

I'll never finish...given the impossibility of anticipating every consequence of the design, after the next revision, more tweaking, theorizing, armchair speculation on human psychology and functorial vomitological barfology with coefficients in metrized navel lint modules over graded earwax co-algebras will have taken us past the point of diminishing returns, long after the train has left the abandoned station and the ostrich has finally taken off, at one minute after midnight on the Doomsday Clock. The plan now is to make these few changes, and if we're lucky, to bootstrap our way into some understanding of what we've done. Not to preclude iterating on the design when some or all of my assumptions turn out to be breathtakingly stupid.

I guess Keyboard Maestro doesn't make it easy to parse

I'm Macless, except for a 2007 iMac, which only runs an older version of The Archive. I haven't messed with Keyboard Maestro. You got me there. I'm mostly submerged in Bolgia Two of the Eighth Circle of Windows 10 Hell, in the iconography of Dante's Inferno.

My format is written for Zettlr. I use minimal YAML. I don't want YAMLing Zettels.

Following the 𝖅𝖊𝖙𝖙𝖊𝖑-𝕿𝖗𝖆𝖓𝖘𝖋𝖔𝖗𝖒𝖆𝖙𝖎𝖔𝖓𝖕𝖗𝖎𝖓𝖟𝖎𝖕 to reformulate what you've read in your own words, I read @Sascha to say that a hashtag that is specific to the Zettel in which it appears has greater utility than a hashtag that is less specific to the Zettel in which it appears. On the spectrum from concrete to sublimely abstract, choose cement. I could replace the phrase "Keywords should be object tags, not category tags" with "Keywords should be specific to the content of

`immutableID`

."GitHub. Erdős #2. CC BY-SA 4.0.

Apologies for another TL;DR. I'm going to maintain the Zettlr Zettel template on GitHub. After a quick review of

Efficiency in learning: evidence-based guidelines to manage cognitive load, it's not immediately obvious that extraneous cognitive load ought to be eliminated in every case (this is relative), though generally it is additive. Neither is it obvious that the sole function of the ZK is the creation of schema (it could be one function) or that "atomic note" should be interpreted as a "large chunk" or a schema. This gets us into instructional design. "Processing"Efficiency in learning: evidence-based guidelines to manage cognitive loadin the ZK using the Zettel format is likely to be a good test.GitHub. Erdős #2. CC BY-SA 4.0.

I ran out of time before I could include the reference (Clark, Nguyen, and Sweller 2006) below on Cognitive Load Theory.

Nguyen, F., Clark, R. C., Sweller, J. (2006). Efficiency in Learning: Evidence-Based Guidelines to Manage Cognitive Load. Germany: Wiley.

Once again I've spoken too soon. Some humility is in order. In (Brünken, Plass, and Moreno 2010) there are several effects to consider. The additive load hypothesis states that cognitive load is the sum of intrinsic, germane and extraneous cognitive load. There is the expertise reversal effect. The worked example effect. It's not at all obvious how to relate Zettel design and ZK to CLT.

Brünken, R., Plass, J. L., Moreno, R. (2010). Cognitive Load Theory: Theory and Applications. United Kingdom: Cambridge University Press.

How is that helpful? Does the Google search engine have any understanding of "units of thought"? If a symbol is a unit of thought, do you have examples of "atomic" Zettels that contain a single symbol in the body?

My point was that there is no useful measurable standard. Even going by cognitive load, might count as atomic for a novice might be "sub-atomic" and counterproductive for an expert, leading to the expertise reversal effect if the novice and the expert happen to be the same person at different times.

GitHub. Erdős #2. CC BY-SA 4.0.

## 20211116105404 Zettel format v2

#zettel #zettelkasten #ruleofthrees #folgezettel #keywords

RELATED[[20211115172141]] Zettel format: revised,[[20210424152745]] Zettel format: rule of threes,

[[20210424174054]] Zettlr title format #+ImmutableID+Title,

[[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

Revision v2 following (Simpson 2021).

`timestamp.md`

in my implementation.## 1. Header: in three plus 1 parts

`immutableID`

, followed by a title, referred to in this Zettel by`title`

;RELATEDfollowed by a list of IDs of prior Zettels related to`immutableID`

; see 1.c.2 below.## 1.a YAML frontmatter: optional

---

reference-section-title: References

---

## 1.b. An immutableID and title at heading level 1

#

`immutableID`

`title`

## 1.c.1. Keywords in #hashtag format

#keyword #example

## 1.c.2. RELATED Zettel IDs

The keyword

RELATEDfollowed by a comma-separated list of Zettel IDs such that for each`ID`

in the list,`immutableID`

either:`ID`

;`ID`

; or,`ID`

.## 2. Body: an atomic note

`immutableID`

is added toRELATED.## 3. Footer: References

## References

Ahrens, Sönke. 2017.

How to take smart notes: one simple technique to. https://www.overdrive.com/search?q=B41A3269-BC2A-4497-8C71-0A3F1FA3C694.boost writing, learning and thinking - for students, academics and nonfiction book writers

anonymous. 2021. “Literature Notes, Where Do They Go Once They Become Permanent Notes?”

Zettelkasten Forum(blog). March 26, 2021. https://forum.zettelkasten.de/discussion/1749/literature-notes-where-do-they-go-once-they-become-permanent-notesChakraborty, Mihir Kumar, Soma Dutta, and SpringerLink (Online service).

2019.

Theory of Graded Consequence A General Framework for Logics of.Uncertainty

Clark, Ruth Colvin, Frank Nguyen, and John Sweller. 2006.

Efficiency in learning: evidence-based guidelines to manage cognitive load. Essential resources for training and HR professionals. San Francisco, CA: Pfeiffer, a Wiley imprint.Doctorow, Cory. 2009. “Cory Doctorow: Writing in the Age of Distraction.” January 7, 2009. http://www.locusmag.com/Features/2009/01/cory-doctorow-writing-in-age-of.html

Fast, Sascha. 2018. “The Difference Between Good and Bad Tags.” Blog.

Zettelkasten(blog). September 24, 2018. https://zettelkasten.de/posts/object-tags-vs-topic-tagsFast, Sascha. 2021. “Write for Your Future Self.” July 29, 2021. https://forum.zettelkasten.de/discussion/comment/12480/#Comment_12480

Gawande, Atul. 2010.

The checklist manifesto: how to get things right. New York, N.Y: Metropolitan Books.Moeller, Hans-Georg. 2012.

The radical Luhmann. New York: Columbia University Press.Plass, Jan L., Roxana Moreno, and Roland Brünken, eds. 2010.

Cognitive load theory. 1. publ. Cambridge: Cambridge University Press.Simpson, Will. 2021. “Comment 13603 on Zettel Format.” Blog.

Zettelkasten Forum(blog). November 15, 2021. https://forum.zettelkasten.de/discussion/comment/13603/#Comment_13603Williams, Joseph M., and Joseph Bizup. 2017.

Style: lessons in clarity and grace. Twelfth edition. Always learning. Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris: Pearson.GitHub. Erdős #2. CC BY-SA 4.0.

Great idea. I'd love to see what you come up with.

AND

I'm a neophyte with peripheral experience thinking about the effects personally of cognitive load. This topic, particularly the book Efficiency in Learning: Evidence-Based Guidelines to Manage Cognitive Load, has ignited a blooming drive. I've gotten this ebook, and the TOC looks inviting.

This appears to be written with pedagogy in mind, but it is easy to slip

instruction to selfinto the mix.Great idea! Thanks.

kestrelcreek.com

Your template is better than great. Thinking through the activity of zettelkasting is both fun and gratifying. Your thinking/writing is much finer detailed, and reasoned than my thinking. I'm a Clod-Luddite stumbling through what little time I have left. By your example, you are setting the bar a little higher and pushing me to rethink.

YMMV, I found over time, using the practice of 'just' listing related zettel, either at the top or bottom of a zettel, became too simplistic. Simplistic in terms of meaning. The list became hard to decipher, mainly if it contained more than a couple of links. Some links in this section were clear about the relationship, and some were not, requiring deeper investigation. Of late, I try to put all my links interstitially within the body of the zettel or use a notation structure like the example in the snippet below, where I add a comment under each link's title UUID`. I found this practice helps my future self. **I'm working on the discipline required to follow through with this. **

kestrelcreek.com

Thank you.

I have mismanaged the cognitive load. I think it's worth looking into CLT. I have too many links.

In the Github version, which I intend to create today, I was going to point out that keywords like

RELATEDin the template have the meaning spelled out in the template, and not the usual dictionary meaning. However, the tendency not to adhere to definitions is so strong a human impulse (I didn't adhere to it) that I'm going to use a term from graph theory instead:ADJACENT. I could have usedFOLGEZETTEL, which would be close to my intention, but I won't. What was my intention? To adapt Folgezettel to digital ZK, with timestamps, for example, in a useful way. Folgezettel worked for Luhmann. I'm not yet ready to throw out the analog baby with the analog bathwater (only to have to change the diapers of a baby android).The idea was to find a small set of neighboring Zettels-- ones that

`immutableID`

directly continues; or that comment on (or raise a question about) an aspect of`immutableID`

. I violated this in my example--I will remove a couple. If I add them to the body, then they have to go into the GitHub repository as example Zettels. ByADJACENTI mean those, and only those (maybe I'll give a little and forward link to the immediate follower).ADJACENTmeans those closest to`immutableID`

, not just related.Too bad, I like

RELATEDbetter.What should go into the GitHub repository?

If this is any good, others will fork the repository and it adapt for their software, make fun of my choices, maybe come up with variations worth incorporating...or it will sit there moribund.

Note to self: in support of "Avoid

The Elements of Style" in the template remarks (3. Body), add the citation (Pullum, “The Land of the Free and The Elements of Style,” English Today 102, vol. 26, no. 2 [June 2010], introduction.)Pullum, Geoffrey K. The Land of the Free and

The Elements of Style. English Today 102 (vol. 26, no. 2, June 2010) http://www.lel.ed.ac.uk/~gpullum/LandOfTheFree.pdfAlso

Pullum, Geoffrey K. Fear and Loathing of the English Passive. Language and Communication 102 (vol. 26, no. 2, June 2010), 34-44. http://www.lel.ed.ac.uk/~gpullum/passive_loathing.html

GitHub. Erdős #2. CC BY-SA 4.0.

@Will @ctietze: the Zettel template is online at https://github.com/flengyel/Zettel.

The keyword

RELATEDwas replaced withCONTEXT. I added an updated version of the template to the README.md file itself. The original markdown file for the template is slightly behind. This will have to be updated.GitHub. Erdős #2. CC BY-SA 4.0.

A version stripped down to essentials would be better to start with. It's hard to see the simple structure. There are a couple of variations depending whether the references are present, entered by hand or generated with Zotero.

GitHub. Erdős #2. CC BY-SA 4.0.

An abbreviated template, with a checklist. The checklist Markdown works better on GitHub at https://github.com/flengyel/Zettel and in Zettlr. A revision of the longer Zettel template follows. @Will Simpson made me realize that the keywordRELATEDam biguous, so I replaced this with the less—or at least "differently"—biguous keywordCONTEXT. Apologies for an unambiguously desperate pun.## 20211119115201 Abbreviated Zettel template v2.1

#replace #these

CONTEXT[[20211118010533]] Zettel template v2.1A Zettel has three parts: a header, a body, and a footer. The header starts with an optional YAML header and includes a level 1 (H1) header with a timestamp ID and a title; a list of hashtag keywords; and a CONTEXT line that links to a Zettel that this Zettel continues or comments (or raises a question about) on an aspect of that Zettel (if there such a Zettel). This text is part of the body. Footnotes and endnotes become links within the body to other Zettels. That leaves the References section for the footer. The References section below can be omitted, generated with a references manager such as Zotero and a YAML header [[20211118010533]] Zettel template v2.1; or added by hand, as below.

`timestamp.md`

in my implementation.The following checklist in Markdown is a first attempt (Gawande 2010).

[ ] Replace the link after the

CONTEXTkeyword with a link to a Zettel Z such that this ZettelCONTEXTline entirely.[ ] Replace the body (including this cheklist) with your own text and Markdown in the body.

[ ] Are there footnotes or endnotes?

[ ] If there are literature citations from a citation manager (Zotero is assumed here)

[ ] Add citations in Pandoc format.

[ ] A references section can be added by hand, as below.

## References

Ahrens, Sönke. 2017.

How to take smart notes: one simple technique to boost writing, learning and thinking - for students, academics and nonfiction book writers. https://www.overdrive.com/search?q=B41A3269-BC2A-4497-8C71-0A3F1FA3C694.anonymous. 2021. “Literature Notes, Where Do They Go Once They Become Permanent Notes?”

Zettelkasten Forum(blog). March 26, 2021. https://forum.zettelkasten.de/discussion/1749/literature-notes-where-do-they-go-once-they-become-permanent-notesGawande, Atul. 2010.

The checklist manifesto: how to get things right. New York, N.Y: Metropolitan Books.GitHub. Erdős #2. CC BY-SA 4.0.

## 20211118010533 Zettel template v2.1

#zettel #zettelkasten #ruleofthrees #folgezettel #replace #these

CONTEXT[[20211115172141]] Zettel format: revisedRevision v2.1 following remarks by (Simpson 2021), with additional citations.

`timestamp.md`

in my implementation.## 1. Header: in 3 + 1 parts

`immutableID`

, followed by a title, referred to in this Zettel by`title`

;CONTEXTfollowed by a list of IDs of prior Zettels providing the immediate context for`immutableID`

; see 1.c.2 below.## 1.a YAML frontmatter: optional

---

reference-section-title: References

---

## 1.b. An immutableID and title at heading level 1

#

`immutableID`

`title`

[[20210424174054]] Zettlr title format #+ImmutableID+Title,

## 1.c.1. Keywords in #hashtag format

#keyword #example

## 1.c.2. CONTEXT Zettel IDs

The keyword

CONTEXTfollowed by a comma-separated list of Zettel IDs such that for each`ID`

in the list,`immutableID`

either:`ID`

;`ID`

; or,`ID`

.CONTEXT,The keyword

CONTEXTapplies to Zettel IDs satisfying the above conditions only, including those that might provide context for ```immutableID`` in other senses of the term.## 2. Body: an atomic note

`immutableID`

is added toRELATED.## 3. Footer: References

[[20210803113219]] Zotero: citing BetterBibTeX references in Zettlr

## References

Ahrens, Sönke. 2017.

How to take smart notes: one simple technique to boost writing, learning and thinking - for students, academics and nonfiction book writers. https://www.overdrive.com/search?q=B41A3269-BC2A-4497-8C71-0A3F1FA3C694.anonymous. 2021. “Literature Notes, Where Do They Go Once They Become Permanent Notes?”

Zettelkasten Forum(blog). March 26, 2021. https://forum.zettelkasten.de/discussion/1749/literature-notes-where-do-they-go-once-they-become-permanent-notesChakraborty, Mihir K., and Soma Dutta. 2019.

Theory of graded consequence: a general framework for logics of uncertainty. Logic in Asia: Studia Logica Library. Singapore: Springer.Clark, Ruth Colvin, Frank Nguyen, and John Sweller. 2006.

Efficiency in learning: evidence-based guidelines to manage cognitive load. Essential resources for training and HR professionals. San Francisco, CA: Pfeiffer, a Wiley imprint.Doctorow, Cory. 2009. “Cory Doctorow: Writing in the Age of Distraction.” January 7, 2009. http://www.locusmag.com/Features/2009/01/cory-doctorow-writing-in-age-of.html

Fast, Sascha. 2018. “The Difference Between Good and Bad Tags.” Blog.

Zettelkasten(blog). September 24, 2018. https://zettelkasten.de/posts/object-tags-vs-topic-tagsFast, Sascha. 2021. “Write for Your Future Self.” July 29, 2021. https://forum.zettelkasten.de/discussion/comment/12480/#Comment_12480

Gawande, Atul. 2010.

The checklist manifesto: how to get things right. New York, N.Y: Metropolitan Books.Moeller, Hans-Georg. 2012.

The radical Luhmann. New York: Columbia University Press.Plass, Jan L., Roxana Moreno, and Roland Brünken, eds. 2010.

Cognitive load theory. 1. publ. Cambridge: Cambridge University Press.Pullum, Geoffrey K. 2010. “The Land of the Free and

The Elements of Style.”English Today26 (2): 34–44. https://doi.org/10.1017/S0266078410000076———. 2014. “Fear and Loathing of the English Passive.”

Language & Communication37 (July): 60–74. https://doi.org/10.1016/j.langcom.2013.08.009Simpson, Will. 2021. “Comment 13603 on Zettel Format.” Blog.

Zettelkasten Forum(blog). November 15, 2021. https://forum.zettelkasten.de/discussion/comment/13603/#Comment_13603Williams, Joseph M., and Joseph Bizup. 2017.

Style: lessons in clarity and grace. Twelfth edition. Always learning. Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris: Pearson.**GitHub. Erdős #2. CC BY-SA 4.0.

@ZettelDistraction, thanks for sharing your ideas about a checklist. Below is my note on this topic with revisions spurred by your example.

I am thinking in the context of a simplified workflow. I create notes exclusively using the templating the software stack I use allows. It provides an algorithmically based choice for some of the items on the checklist based on the template choice. Because the software handles them, it seems supercilious to have them on the checklist.

What I record for the title, subtitle

^{1}, and tags are place-holders initially. They best-workedduring, and afterthe note is drafted. This circular feedback makes the structure/order of the checklist items a bit hard to formalize.[edit]

My workflow/practice is evolving towards placing tags and links interstitially within the note. This is a more fine-grained approach than placing them all together.

[/edit]

## Zettel Crafting Checklist 202110110727

---

UUID: ›[[202110110727]]

cdate: 10-11-2021 07:27 AM

tags: #zettelkasting #checklist

---

## Zettel Crafting Checklist v.2

If surgeons and airline pilots can benefit from checklists, they probably will help me.^{2}It is essential to keep any formal checklist flexible and straightforward.

If every zettel is an exception, it points to a failure of the checklist to provide guidance.

There is a difference between the structure of a zettel and its content.

The checklist guides the structure but only hints/prompts the content.

My idea of a zettel-making-checklist is that it focuses on the structure, thereby freeing up cognitive cycles so I can focus on the zettel content. And heaven knows I need all the cognitive cycles I can muster.

New NoteYAML Frontmatter [[202003231450]]

[ ] UUID present? (Provided by the software preferences.)

[ ] Creation date and time present? (Provided by the software preferences.)

[ ] Tags present? (Added and revised during or after the body is created.)

Body

[ ] Title present? (Reconsider title as note develops. The title wants to be short, concise, and relevant. It wants to be descriptive of the content.)

[ ] Is a one or two summary sentence present at the head of the note. It should answer the question, "What is the central/essential thing to remember?" (Best done after the note is developed in draft one and continually revisited.)

Actual zettel (may include one or more of the following)

[ ] Connect to an existing structure note or make this the first zettel connection in a new structure note?

[ ] Ask if deep links are present? Look for connections with keyword and key idea searches.

[ ] Is all #beautiful language captured and tagged

Footer/References/Citations/NOtes

[ ] Footnotes

[ ] Bibliographic citations as needed?

[ ] Are they present in Zotero?

[ ] URLS (web, Evernote, Bear)

This is only the first draft. Refactor, Refactor, Refactor!Is a one or two summary sentence present answering the question, What is the central/essential thing to remember? ↩︎

http://atulgawande.com/book/the-checklist-manifesto/ ;↩︎

kestrelcreek.com

Our workflows and formats are different enough that at most I will borrow the idea to add a step at the end to take a second pass at the title, keywords, etc., and perhaps add a summary sentence.

I think my checklist and format attempts to separate concerns...

The formats and checklist are on GitHub at https://github.com/flengyel/Zettel

I could add the formats and checklists you and others use there. It might be helpful to start a repository of Zettel formats, checklists and workflows that different people use. Examples can be helpful.

Your signature has been updated:

"I'm a futzing, second-guessing, backtracking, compulsive oversharing, ZK-maniac, in other words, your typical zettelnant."

GitHub. Erdős #2. CC BY-SA 4.0.