Mathematical definition of folgezettel and operatic lyricism 202110010741

I'm still wondering what is so fascinating about your mathematical dissection of the folgezettel. This morning it struck me musically. The symbology is the music, and your vocabulary being like operatic lyrics. First, you take one tack, then you quickly move to another. Sometimes quick and suddenly lingering on a detail.

I bounce in and out of your stream of proof. It is fast-moving, and I'm floating along up to my neck, getting glimpses of insight now and then as my head bobs along.

ps. I'm stealing your brilliant idea of posting some #forum-posts as zettel. I can draft them in The Archive, which is a more comfortable writing and thinking environment.

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon 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:
...
ps. I'm stealing your brilliant idea of posting some #forum-posts as zettel. I can draft them in The Archive, which is a more comfortable writing and thinking environment.

Thank you. I too have borrowed your discovery that keywords work in the forum. With posts in Zettelform, an organic development, the forum could itself become a collaborative digital Zettelkasten—if the hosts don't object.

Users would see different Zettel formats in action, and their possibilities. Over time it could become what Niklas Luhmann called a "communication partner." This phrase seems epistemically loaded: taking Luhmann at his word, it suggests that Luhmann and the Zettelkasten achieved common knowledge. Luhmann knew that the slip box knew what Luhmann added to it, and the slip box knew that Luhmann knew that the slip box knew, and so on. Perhaps the online digital Zettelkasten could achieve what virtual public spaces so far cannot do, which is to foster common knowledge, as opposed to distributed information.

Borrowing an idea from identity management, I referenced your post as "202110010741@Will," which will suffice here, though not in a global distributed Zettelkasten...

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

This must be the funniest joke in the History of Zettelkasten Science. What a strange loop you are.

I think I missed the beginning, so: What are you aiming at?

Can I assist you with non-math stuff?

TL;DR: for 1. emulate Luhmann's system in a digital ZK; augment Zettel format to help track links; write update rules; write down precise mathematical definition of Folgezettel, partly to illustrate the difference between verbal theories and mathematics; write up a specific implementation "Digital Zettelkasten Step-by-Step." Need to review Schmidt.

For 2, still thinking...I am probably misreading Luhmann, or I haven't read enough.

A. [skip to B] Ultimately, I don't know what I am aiming at. I am driven to aim at something. [I say driven instead of the word passion, a word employers have come to expect to hear from employees describing their attitude toward their employment, whatever their working conditions.] God willing, that something I happen to be aiming for does exist and is worth aiming for; or it doesn't exist, but I have neverthlesss aimed close enough in the direction of something that does exist and that is worth aiming for; or it doesn't exist, there is nothing close enough that is worth aiming for, but the result of pursuing something nonexistent was worth the effort despite my abysmal ignorance.
A/1. Whatever reasons I might give for aiming at something might or might not be acceptable as reasons. Philosophers have never agreed on what counts as a reason in the 2500 year history of philosophy.
A/2. Then again, I could be ineffective in ways that a more experienced practitioner could identify.
B. But originally, I wanted to develop a checklist for working with Zettelkasten.
B/1. And I wanted to enhance the Zettel format I use, if necessary.
C. Niklas Luhmann was a sociologist who thought in terms of systems.
D. He developed and refined his Zettelkasten as a system for research and for thinking generally.
E. I wanted to see if Luhmann's system could inform my efforts, using a digital ZK.
E/1. I myself am interested in some of the applied mathematics of sociology developed since at least the middle of the 20th century, as this applies to some sociological problems of importance.
F. Luhmann's system relied on a numbering scheme that others have called Folgezettel.
G. He explains that his system was designed to introduce an element of surprise into his thinking, a middle ground between predictable order and utter chaos.
H. Since he regarded his ZK as a "communication partner," he had to surprise it. And it had to surprise him.
J. There are various mathematical characterizations of random processes that fall between predictable order and chaos.^{[CITATION NEEDED]} There are also measures of surprisal, Kullback–Leibler divergence $(\displaystyle D_{\text{KL}})$ (also called relative entropy). But we are far from measuring this in a Zettelkasten.
K. Luhmann claimed that his note numbering scheme, together with internal linkings of notes, worked to introduce a useful, exploitable type of randomness that would facilitate further thinking and writing. Enough to earn his keep as an academic—more than enough, even today
L. Encoded in the numbering scheme is the decision to introduce notes that continue a line of thought, that introduce a side-thought, or that start a new thought altogether.
M. Verbal descriptions of processes are likely to suffer from ambiguity. In [Gobet, F., Lane, P. C., & Lloyd-Kelly, M. (2015). Chunks, Schemata, and Retrieval Structures: Past and Current Computational Models. Frontiers in psychology, 6, 1785] Gobet et al write, "...not enough constraints are provided by verbal theories, and thus too much freedom is left in the way they can be interpreted."
N. I thought that I needed a mathematical description of Folgezttel to overcome the limitations of verbal theories.
O. One might object that Folgezettel are so simple that a verbal description is enough to avoid ambiguity.
P. But the Internet shows considerable back and forth over the significance of Folgezettel. And a verbal description isn't a programmable formula. The main issue is whether Luhmann's design did provide the middle ground between predictable order and chaos.
Q. A mathematical description will provide something to reason with.
R. Links can emulate Folgezettel--that is, links can express the same adjacency relations between notes as Folgezettel, but they do not record the decision that was made at the time a note was initially added to a ZK. This has to be added separately--by adding to the Zettel format, for example.
R/1 Links can emulate total disorganization or predictable order. They are unconstrained without a process.
S. Maybe the random graphs of probabilistic combinatorics are enough. Again there is the question of measurement and procedure.
T. Structure notes have been proposed as an alternative to Folgezettel. The question then is when to add them.
U. We are still far from characterizing the kinds of graphs and procedures likely to facilitate "productive" effort. This is assuming there is a combination of them that have some productive effect. The null hypothesis is that no graph configuration or procedure for extending a graph has any measurable effect.
V. But we have nothing to calculate with, nothing to measure the "semantic distance" between notes (I'll get back to you--maybe work of Bob Coecke and others in the UK is relevant), as this played a role in Luhmann's process. Only verbal theories.
V/1. Not that working systems require a theory. Most systems are under-explained and continue to function without the system realizing that no one understands how it works. The (macro-)economy is one such system.
W. Sorry for being such a wet blanket.
X. Still, I wanted a description for myself, with a comprehensible system for producing a ZK along Luhmann's lines, one that could readily be translated into a digital ZK.
Y. The linearization is an easily programmable map--it's a substitution.
Z.And I wanted to push the notion of Folgezettel somewhat, and for that, I needed mathematical definitions.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

Comrades, fellow Zettlers, Zettelscenti, tovarishch Commissars: you missed an opportunity to draw attention to a spectacular public screw up. The generalized Folgezettel numbers survive, there are linearizations, but $(L)$ is too good to be true.

Counterexample. The map $(L)$ is surjective but not injective.
$$(
L\left(1\mathbf{.}2\mathbf{.}3\right)=L\left(1\left.\right|_2 3\right)=
1\mathbf{.}2\mathbf{.}3.
)$$

There are many $(L)$-preimages of $(a.b.c.d)$, for example,
$$(
a.b.c.d, a|_b c.d, a.b |_c d
)$$ are preimages.

The map $(L)$ should be redefined (on finite subsets) to be order-preserving and injective. A question of shifting images out of the way. It's always possible to prepend a large enough integer to the value...but unless this is done carefully, it won't reproduce the "inner expansion" of Luhmann's system.

Linearization

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

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

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

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

The map $(L)$ immediately generalizes to IDs generated by any set

No, you're mistaken. Time for more "story editing."

Story editing and deflection. People make errors like this all the time, with the exception of historical figures like Gauss, Newton, Noether, John von Neumann, Erdös, Grothendieck, Gödel, and of course, Richard Stanley. Besides, a linearization isn't needed in the digital case.

Rejoinder of the Default Mode Network. This should have been obvious to anyone, even at 3 AM in the morning. Your errata [sic] for $(L)$ was wrong! Ouch! Own up: you can't do it. Or stop stalling and fix it. You can't let Luhmann and the Folgezettel down now.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

POZZO It's a disgrace! But there you are.
—Samuel Beckett. Waiting for Godot.

It helps to calculate with paper and pencil on occasion, instead of "calculating" (read: typesetting) directly on the screen in LaTeX. The idea is to force descendants of $(w)$ to appear one after the other, in order of the branch number $(n)$, taking advantage of the ordering of strings of zeros. The successor of $(w)$ will appear after the descendants of $(w)$. Illustrate this.

Linearization

Proposition. There is an injective, order-preserving map $(L)$ from the partially ordered set of Folgezettel IDs to the lexicographically ordered set of nonzero decimals. $$(L: \left(\mathcal{F}(\mathcal{D}), \preceq\right)\rightarrow \left(\mathcal{D}_{\ne0}, \preceq\right))$$

Proof. The map $(L)$ is defined inductively 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 $(w\in \mathcal{F})$.

Complete the proof.

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

Let's get a monoid homomorphism out of this. The map $(L)$ is the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols, pre-composed with the inclusion of $(\mathcal{F(\mathcal{D})})$ in $(\Sigma^*)$, and co-restricted to $(\mathcal{D}_{\ne0})$. No one writes like this—not even an academic. Say instead that $(L)$ can be obtained from the unique lift of the following map on symbols. Or define the lift and then the (co)restrictions.
$$(L(w) =
\begin{cases}
w& \text{if}\quad w\in\mathbb{Z}^+;\\
\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}&\text{if}\quad\exists n\in\mathbb{Z}^+, w= `\left.\right|_n\text{'};\\
\mathbf{.} &\text{if}\quad w=`\mathbf{.}\text{'}.
\end{cases} )$$ for $(w\in \Sigma)$. Why do we care about a monoid homomorphism? Because computers. String substitution algorithms are fast.

Now use the linearization map $(L)$ to say what the linear ordering in the slip box is. It would help to give an honest proof. 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 $(w,x\in\mathcal{F(\mathcal{D})}.)$

I'll sleep on this...

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

Take the domain of $(L)$ to be $(\mathcal{D}^+)$. Then the $(L)$-image of a positive Folgezettel ID $(w)$ is positive only if $(w)$ is a positive decimal. Otherwise $(L(w))$ is a normalized non-positive decimal. The codomain of $(L)$ is the set of normalized decimals, and the map is injective. The three cases are clear. As for order-preserving, write this out. Either $(u, v)$ are on the same branch (clear), or one is a descendant of the other. The ancestor will be smaller than the descendant--this holds for $(L)$-images too. Rewrite the preceding fumble with proofs.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

20211003001710 The non-linear road to linearization

Well. No shit. —R. Lee Ermey. Full Metal Jacket.

Pushing the definition of the generalized Folgezettel IDs as far as possible to obtain a simple calculational linearization requires the strong condition that all the terms of the ID are positive. The normalized Folgezettel IDs don't have a nice linearization, and we can forget about using the denseness of the nonzero decimals to insert notes in between pre-existing notes. This positivity constraint forces us to keep going forward—a positive development, if you allow me a non-mathematical remark. But positive generalized Folgezettel IDs are good enough.

Partial order

The proof that linearization is order-preserving needed an inductive definition of the partial order, as follows. We should go back and make some minor adjustments, but everything goes through. Come back tomorrow! Pay no attention to the man behind the terminal...

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

Factlet. 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 $(L)$ from the partially-ordered set of positive Folgezettel IDs to the lexicographically ordered set of normalized decimals. $$(L: \left(\mathcal{F}(\mathcal{D}^+), \preceq\right)\rightarrow \left(\mathcal{D}, \preceq\right))$$

Proof. Let $(\mathcal{F}^+ = \mathcal{F}(\mathcal{D}^+).)$ (This is previous notation.) 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 $(w\in \mathcal{F}^+)$.

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

Edited again by @ZettelDistraction. Sorry about that @ctietze. In case an unqualified apology and a promise not to let it happen again doesn't cut it, I'll work something out.

$(L)$ 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 $(x\in\mathcal{F}^+,n\in\mathbb{Z}^+, d\in\mathcal{D}^+.)$
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 parameterized induction hypothesis that for each $(r\in\mathcal{F}^+)$, whenever $(s\in\mathcal{F}^+, s < t)$, $(L(r) = L(s) \Rightarrow r = s)$.

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 (this is mathematics not identity politics, nevertheless Jesus Christ, now the "integer positive" decimals are getting 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 $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\left.\right|_n d = L(x\left.\right|_n d),)$$ 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, we said it. Explicitly. There are three cases.

Case 1. The case of $(w\in\mathcal{D}^+, x\prec y)$ 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

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

Let's get a monoid homomorphism out of this. 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 $(w\in \Sigma)$. Why do we care about a monoid homomorphism? Because computers. The string substitution is $(O(n))$ in the length $(|n|)$ of the ID.

The linearization map $(L)$ is used to define the linear ordering in the slip box. 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 $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

The observation that if you think you're done, you're only 40% done applies. This "process" bears a greater resemblance to an episode from the first season of The Outer Limits (1963) than it bears to research. But it is both. Do not attempt to adjust your web browser. You are about to participate in a great adventure. You are about to experience the awe and mystery that reaches from the inner mind toThe Outer Zettelkasten.

It's not professional to overlook cases. The full cases are

The proof that linearization is order-preserving needed an inductive definition of the partial order, as follows. We should go back and make some additions, but everything goes through.

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

Factlet. 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 $(L)$ from the partially-ordered set of positive Folgezettel IDs to the lexicographically ordered set of normalized decimals. $$(L: \left(\mathcal{F}(\mathcal{D}^+), \preceq\right)\rightarrow \left(\mathcal{D}, \preceq\right))$$

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

The sociologist Niklas Luhman assigned identifiers (IDs) to Zettels within his Zettelkasten to maintain, within the linear ordering of the Zettelkasten, a tree structure that reflected semantic relationships among nearby Zettels, and that possessed an internal branching property.

We give a mathematical formalization of Niklas Luhmann's unique, immutable Zettel IDs, sometimes referred to as Folgezettel. The partially ordered set of generalized Folgezettel IDs is defined first, 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, called a linearization map, 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 that 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 a calculational linear-time 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 given 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

$$(0\prec 0\mathbf{.}0\prec 0\mathbf{.}0\mathbf{.}0\mathbf{.}0\mathbf{.}0 \prec 0\mathbf{.}1\prec 0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\prec 1\mathbf{.}0\mathbf{.}0\mathbf{.}1 \prec 2\mathbf{.}0\prec 2\mathbf{.}0\mathbf{.}0)$$ The nonzero condition eliminates "infinitesimals," which are decimals where every digit is zero. 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 $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k\in\mathcal{F}, )$ $(w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m \in\mathcal{F})$,
$$(\left(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k = w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m\right) \Leftrightarrow \left(k=m \land i_n =j_n\, \land v_n=w_n\right))$$ for $(1\le n\le m\text{.})$

NOTE: rather than make a heroic overnight effort to complete this in one sitting using previous notes in this thread (to scoop the brutal, high-stakes competition in this 1000 atmosphere pressure-cooker of a field), I'm going to sleep. The inductive definition of $(\mathcal{F})$, the inductive definition of the partial order $(\preceq_{\mathcal{F}})$, the inductive definition of the linearization map $(L)$, the proof that $(L)$ is an order-preserving bijection and the lift of $(L)$ to a monoid homomorphism (etc) will be written up shortly in subsequent posts.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

NOTE: Continuation of 20211004235454Formalization and Generalization of Niklas Luhmann's Folgezettel IDs 1. Because the preview feature doesn't render MathJax, I'm going to stop here and review the Linearization argument etc., tomorrow (late again—bedtime). Caught a typo in 1: we will be concerned with $(\mathcal{D}^+)$ and $(\mathcal{D})$, respectively. The sets of positive and normalized decimals were reversed.

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 $(`\left.\right|_1\text{'})$. (We assume that we can eliminate definitions efficiently [cite Elimination of Definitions and Skolem Functions in First-order Logic, by Jeremy Avigad, if this isn't overkill].) An ID $(w\in\mathcal{F})$ is a Luhmann ID if $$(
w= \begin{cases}
d, &d\in\mathcal{D}^+;\\
d_1 /\cdots / d_k, &\exists k, \in\mathbb{Z}^+, d_j\in\mathcal{D}^+, 1\le j\le k.
\end{cases})$$ The Luhmann IDs allow a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.

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

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 $(w\in \mathcal{F})$.

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

Edited again by @ZettelDistraction. Sorry about that @ctietze. In case an unqualified apology and a promise not to let it happen again doesn't cut it, I'll work something out.

$(L)$ 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 $(x\in\mathcal{F},n\in\mathbb{Z}^+, d\in\mathcal{D}^+.)$
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 $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\left.\right|_n d = L(x\left.\right|_n d),)$$ 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, we said it. Explicitly. There are three cases.

Case 1. The case of $(w\in\mathcal{D}^+, x\prec y)$ 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

The map $(L)$ extends to 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.

The linearization map $(L)$ is used to define the linear ordering in the slip box. 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 $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

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.

Concluding Unsympathetic Postscript

Will fix this heading also

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

It would be helpful to produce diagrams, and program a few utilities to recognize positive and normalized decimals and positive Folgezttel IDs; to transform a positive ID to a normalized decimal; and to compare two decimals in the lexicographic order. These could be used to help illustrate the inner ramifications of Luhmann's system as notes are added. To that end, here are a few regular expressions for decimals and IDs.

Errata.The last equation of 20211006000400 in the proof of surjectivity of $(L)$ should be $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}d =
L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}d = L\left(x\left.\right|_n d\right).
)$$

NOTE: working on regular expressions means that I am uninspired and can barely stay awake.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

B:You wanted to develop a checklist. What kind of checklist?

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.

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)

Another example: An argument has a conclusion. But this conclusion could be the premise for another claim as conclusion. Each type of argument could be asigned a value (deductive or inductive)

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)

Example: Fasting as part of what Chakravarty calls the feast-famine-cycle:

It can be shown in the evolutionary history in various stages (back to our existence in the bacterial stage)

It is practiced abundandly in religious communities (especially for purification)

It is present in classical texts (Greek doctors for example or stoics)

It is present in contemporary empirical research (AMPK, Sirt-1, Insuline Sensitivity, fasting protocols, effects of ghrelin and glucagon, etc.)

It is present in the platonic world (e.g. the co-dependency of Ying and Yang, some theoretical thinking on the nature of hunger -- by me, though)

Strength of evidence and Status of Lindytest are both susceptible to formalisation. I personally use rather primitive means. But perhaps, you can come up with something more sophisticated.

@Sascha said: B:You wanted to develop a checklist. What kind of checklist?

A checklist for working with the ZK.
My Zettel format is getting pretty close to yours.

Example:

Before beginning
[ ] this is a new note
[ ] this is a revision

Header
[ ] # + ID + Title present?
[ ] keywords present?
[ ] CONTEXT present?
Is this note
1. [ ] a continuation of a pre-existing note?
2. [ ] related to a pre-existing note, but not a continuation? (a footnote, endnote, an aside, a digression, another tiresome rant...)
3. A note starting another subject.
4. The immediate predecessor of another continuing this one?

Depending on the answer to 1--4:
Add links to context (I have to figure out the format. I want to follow Luhmann's practice of "link locally, think globally.")

Body:
[ ] are bibliographic citations needed?
[ ] are they present in Zotero?
If so, add them. If not, write TK and keep writing. Writing is not research and conversely. -- Cory Doctorow.
[ ] YAML header for the bibliography present? (I'm using Zettlr. This is my only concession to YAML--so far)

Footer
[ ] Footnotes present? (These could be links to other notes.)

The bibliography is controlled by the YAML header. No need to add a heading for this.

No doubt I have omitted a great deal, such as the obligatory

Caution: failure to adhere to this checklist subjects the non-adherent to a $10,000 fine.

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.

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...

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: An argument has a conclusion. But this conclusion could be the premise for another claim as conclusion. Each type of argument could be asigned a value (deductive or inductive)

Classifying types of arguments is useful....

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.

Well, I am at a temporary impasse.

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

@ZettelDistraction said:
Zettelkasten are so idiosyncratic that it's difficult to know what to formalize.

I'm not so sure this is true. Sure, you and I are different ("so idiosyncratic"). The content and our workflow will differ conceptually. If we both are making a "zettelkasten," then formalizing the conceptual procedures with a simple checklist might be hard but shouldn't be impossible. I don't want to make the mistake of thinking that I'm so uniquely different and separate from the rest of humanity that I have something no one else can understand or help.

If surgeons and airline pilots can benefit from checklists, they probably will help me. ^{1}

Here is my first go at a checklist for new zettel creation.
The Zettel Checklist below follows my current workflow. I'm open to revisions.
I think it 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.

Zettel Checklist

New Note

YAML Frontmatter
[ ] UUID present?
[ ] Creation date and time present?
[ ] Tags present?

Body
[ ] Title present?
[ ] Is a one or two summary sentence present answering the question, "What is the central/essential thing to remember?

Actual zettel (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 that I need to ...

[ ] Connection to existing structure note or the first zettel in a new structure note?
[ ] Deep links present?
[ ] Beautiful language captured and tagged

Structure notes are the digital replacement for [note relationship]. 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...

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.

Structure notes show how zettel relate to each other. My workflow is to put every note on a structure note. The structure note is one of the contexts of the note, an important one. When reviewing a note and I hit on its structure note, I see how I visualized its context and have other zettel listed in the structure note with which I can wander and wonder.

@Sascha said:
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 classic texts

It is present in contemporary empirical research

It is present in the platonic world (or theory)

These, rephrased as questions, are candidates to additions/reminders/prompts in a Zettel Checklist. Thanks @Sascha

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon 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:
Zettelkasten are so idiosyncratic that it's difficult to know what to formalize.

I'm not so sure this is true. Sure, you and I are different ("so idiosyncratic"). The content and our workflow will differ conceptually. If we both are making a "zettelkasten," then formalizing the conceptual procedures with a simple checklist might be hard but shouldn't be impossible.

I was referring to mathematical formalism, not to (nonmathematical) checklists--I offered one of my own in the post you quoted, to make good on the intention to follow Atul Gawande's Checklist Manifesto. Cf. the end of https://forum.zettelkasten.de/discussion/comment/13022/#Comment_13022

@Will If surgeons and airline pilots can benefit from checklists, they probably will help me. [^1]

My thoughts exactly--people's lives are on the line. A single forgotten or misplaced link could be catastrophic.

As in your case, the purpose of the checklist was to ensure the format is consistent and to include some link context (the decision where to link the note with other notes. This is built-in with Folgezettel. In a digital ZK, there is nothing, in the absence of a disciplined practice or a checklist, together with some format specification, to ensure that local connections to nearby notes (the note it continues, comments on, proceeds) are present in a Zettel at a minimum, consistently. You can do anything.

I'll revise my checklist above based on yours. Thank you.

@ZettelDistraction Structure notes are the digital replacement for [note relationship]. 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..

Here I am mistaken. I was thinking of moving the links for the previous and alternative notes, and any successor notes from a context section into a separate note, but this doubles the work without adding anything structure notes don't already provide.

Structure notes show how zettel relate to each other. My workflow is to put every note on a structure note. The structure note is one of the contexts of the note, an important one. When reviewing a note and I hit on its structure note, I see how I visualized its context and have other zettel listed in the structure note with which I can wander and wonder.

By a process of inanition, as opposed to elimination, I cry uncle. If you can't beat 'em, join 'em. I might as well do the same thing. I could still create Folgezettel IDs, for fun..

The generalized Folgezettel IDs define ordinary outlines, outlines within outlines, and so on. The original definition with proofs was written down so that I could get it over with. There were a few missteps for all to see—as of now I am aware of two typos that remain to be corrected, but no serious errors. Perhaps @thomasteepe could look at it. @ctietze mentioned that there were no illustrations to make this more user friendly.

Examples

What do the generalized IDs look like?

Initially, notes are given outline numbers (though they can start anywhere).

Under $(2.1/_1 1)$ you could have other outlines
$$(\begin{array}{}
2.1/_1 1.1 \\
2.1/_1 1.2 \\
& 2.1/_1 1.2.1 \\
& 2.1/_1 1.2.2 \\
2.1/_1 1.3\\
2.1/_1 2.
\end{array})$$ This last ID was already present in the first.

But what is going on here?

A note with ID $(Z)$ can have any number of parallel outlines, with prefixes as follows.
$$(\begin{array}{}
Z /_1 & \cdots & Z/_n &\cdots
\end{array})$$ Their subsections are numbered with outline decimals; e.g.,
$$(
1, 1.2, 3.5.11.1, \ldots,\,\text{etc}
)$$

This can be read as follows: $(Z/_4 3.5.11.1)$ is note $(3.5.11.1)$ in the $(4)$-th alternative comment on note (with ID) $(Z)$.

Now suppose you wanted to start commentary in outline form on $(Z/_4 3.5.11.1,)$ and you decide that the note for this commentary is the first parallel commentary. You assign $(Z/_4 3.5.11.1 /_1)$ as the prefix to all the notes that comprise the new commentary, each of which have section and subsection numbers (I called them decimals). For example:
$$(\begin{array}{}
Z/_4 3.5.11.1 /_1 1 \\
Z/_4 3.5.11.1 /_1 2 \\
& Z/_4 3.5.11.1 /_1 2.1 \\
& Z/_4 3.5.11.1 /_1 2.2 \\
& &Z/_4 3.5.11.1 /_1 2.2.1
\end{array})$$ This last ID identifies the note for section $(2.2.1)$ of the 1-st comment on section $(3.5.11.1)$ of the 4-th comment on (the note with ID) $(Z)$.

How do Niklas Luhmann's IDs fit into this scheme?

That gives you coordinates of outlines within outlines. Niklas Luhmann went as far as one parallel commentary per note (maybe @Sascha could correct me!). In this scheme (up to relabeling alphabetic characters with numbers) you might have
$$(\begin{array}{}
1 \\
2 \\
& 2.1 \\
& & 2.1/_1 1 \\
& & 2.1/_1 2 \\
& & & & 2.1/_1 2 /_1 1\\
& & & &2.1 /_1 2 /_1 1.1 \cdots \\
3
\end{array})$$ Luhmann would not have used the subscripted slashes $(/_1)$. To make this closer to Luhmann's practice, it would appear like so.
$$(\begin{array}{}
1 \\
2 \\
& 2.1 \\
& & 2.1/ 1 \\
& & 2.1/ 2 \\
& & & & 2.1/ 2 / 1\\
& & & &2.1 / 2 / 1.1 \cdots \\
3
\end{array})$$
We could also substitute some letters for numbers, if we don't run out of letters. Luhmann didn't always strictly adhere to this model, but I think it's close enough, and it captures Luhmann's intentions formally.

Linearization

To answer how to arrange these notes in a one-dimensional slip box, there is a bijective order-preserving map from the general Folgezettel to what I call normalized decimals, in which the initial and final numbers are nonzero. The map is called $(L)$ and it sends $(Z/_n d)$ to $(L(Z)\underbrace{.0.0.\cdots.0.}_{n\, \text{zeros}} d)$
by recursion, and $(L(d) = d)$ for a decimal $(d)$ (all the numbers are positive in $(d)$).

The normalized decimals are lexicographically ordered, so that they can be used to arrange all the notes in linear order. If they were paper notes, this map would allow you to write down the Folgezettel ID of a new note and figure out where it belongs in order in your files. Or if a note referred to another note, you could find that one by applying $(L)$ and comparing its lexicographic value with that of other notes.

We left $(Z)$ undefined but this would be another Folgezettel ID. The expressions could be long, but they are easy to compare in the lexicographic order.

What's missing?

$(\mathbf{1}.)$ An illustration showing how the branching outline diagrams are ordered linearly. This should have one of the outline diagrams above, followed by its linearization.

$(\mathbf{2}.)$ An illustration showing the effect on the one-dimensional diagram when a new note or a comment on a note is added to an existing branching diagram. This should illustrate what Luhmann referred to as "internal ramification."^{[CITATION NEEDED]}

$(\mathbf{3}.)$ Whenever I attempt to apply the Rule of Threes, I forget one-third of them.

$(\mathbf{3/_1 1}.)$ now I remember (this is a comment on $(``\mathbf{3}.")$ in the ID notation, which makes it a form of humor even lower than the pun: the mathematical pun).

$(\mathbf{3/_1 1.1}.)$ In addition to the structure notes suggested by @Sascha and @Will, there is nothing stopping us from assigning Folgezettel to notes. It's more work, since Folgezettel links aren't supported in most software, but nothing is stopping us from searching on a Folgezettel ID such as $(1.3/_4 3.5.11.1 /_1 1)$ or on the prefix $(1.3/_4 3.5.11.1 /_1)$ or the prefix $(1.3/_4 )$, which would give us all of the Folgezettel (and references to those Folgezettel) in the branches of the tree starting with those prefixes.

$(\mathbf{3/_1 1.1/_1 1}.)$ My communication partner of a Zettelkasten complains: "Not so fast human!" One could name (or rename) files to have the Folgezettel IDs, suitably translated from LaTeX to an operating system-compatible filename format.

$(\mathbf{3/_1 1.1/_1 1 /_1 1}.)$
My reply to ZK: yes ZK, technically correct. But give me a break! This workflow is asking for trouble! It complicates what should be a straightforward daily practice.^{1}^{,}^{2} I'd ghost you ZK if in addition to assigning Folgezettel IDs within Zettels (and that's in addition to timestamps, which are also filenames), I had to rename already timestamped files to match the Folgezettel ID. That's a rabbit hole I'm avoiding.

$(\mathbf{3/_1 1.1/_1 2}.)$ Speaking of treating me (your ZK) as a communication partner, it's not a bad idea if one takes a journalistic attitude. The journal to which this (admittedly superficial) attitude is directed is none other than the Reader's Digest of pop psychology, Psychology Today, where we learn that the mind is modular.^{3} Allow me to introduce myself: the voice of your ZK is one of those modules. We will make a virtue out of what Freud called “intrapsychic conflict” (cf. "internal conflict" in "Freud. 2nd Edition" by Jonathan Lear).^{4} From now on, you will switch your module to my module when communicating with me, your ZK. I will switch modules to yours when it is your turn to respond—whether you respond or not.

$(\mathbf{3/_1 1.2}.)$ It's possible that adding such numbers would introduce the elements of surprise that Luhmann reported.^{[CITATION NEEDED]} I'm not assuming that linking every Zettel to an associated structure note doesn't introduce elements of surprise. At least one could compare following timestamp filename links along structure notes with searching on Folgezettel IDs. Not that anyone would take my word for it.

References

Maisel, Eric. The Power of Daily Practice: How Creative and Performing Artists (and Everyone Else) Can Finally Meet Their Goals. United States: New World Library, 2020.^{2}↩︎

I'm more than a little embarrassed to cite a self-help book.^{5} But as far as that goes, the psychologist Eric Maisel is one of a handful of writers in that genre worth reading.^{6}↩︎↩︎

Folgezettel and Digital ID's bring complementary views into a zettels context.

@ZettelDistraction said:
My thoughts exactly--people's lives are on the line. A single forgotten or misplaced link could be catastrophic.

Probably not, but all bets are off in other universes. In some other universes, I'd be the one with the Erdös #2 and you'd be an old man holed up in the mountains of Northern Idaho.

As in your case, the purpose of the checklist was to ensure the format is consistent and to include some link context (the decision where to link the note with other notes. This is built-in with Folgezettel. In a digital ZK, there is nothing, in the absence of a disciplined practice or a checklist, together with some format specification, to ensure that local connections to nearby notes (the note it continues, comments on, proceeds) are present in a Zettel at a minimum, consistently. You can do anything.

Let's leave aside the fact that both a digital Zk and folgezettel numbering absolutely require a "disciplined practice" to succeed. Let's look at what each brings to the table. You have clearly demonstrated that folgezettel numbering inherently brings context to the zettel. But what kind? It brings physicality to the order. An initial placement behind notes, as with the start of a new cluster. Or intermingled within an existing cluster, being a continuation or expansion. What this obscurely calls out is the first initial context for the note. The Digital ZK ID doesn't specifically provide this type of context but does provide unique context. A date stamp is context and can be crucial to some future self. The zettel title is a powerful piece of context impossible to replicate with a folgezettel ID of '25.6.2.2'.

Example IDs - which provides more apparent context?
folgezettel - 25.6.2.2
digital - Folgezettel-Digital ID comparison 202110101626

I say all this not to convince you to cry, uncle. I do think there is something there here. You might be starting to hit on it.

@ZettelDistraction Structure notes are the digital replacement for [note relationship]. 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.

#beautiful-language
I like this "think globally, link locally." Yes, it offers a surprise. It expresses a mature, disciplined practice. A digital ZK with structure notes captures the best of both worlds. The structured note has the relationship of folgezettel IDs and the explicit context of digital ZK IDs.

By a process of inanition, as opposed to elimination, I cry uncle. If you can't beat 'em, join 'em. I might as well do the same thing. I could still create Folgezettel IDs for fun.

There is no reason that a folgezettel ID couldn't be added to the YAML front matter or post-pended to the title.
Don't cry, smile because we got to here.

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

The diagrams in "20211010190300 User-friendly diagrams, take one" needed a convention to illustrate parallel outlines. The convention, which I forgot to use, is that parallel outlines are represented in order across, and outlines with the same prefix are represented in order vertically, but without indenting. A shift to a parallel outline below a given Folgezettel ID will be indented. For definiteness the first note of a parallel outline ID starts with the decimal $(1)$, but this isn't required for the system to work.

@Will said:
Let's leave aside the fact that both a digital Zk and folgezettel numbering absolutely require a "disciplined practice" to succeed. Let's look at what each brings to the table. You have clearly demonstrated that folgezettel numbering inherently brings context to the zettel. But what kind? It brings physicality to the order. An initial placement behind notes, as with the start of a new cluster. Or intermingled within an existing cluster, being a continuation or expansion. What this obscurely calls out is the first initial context for the note.

I thought I was explicit in several posts that the Folgezettel ID calls out the initial context for the note. The format enables you to walk up the tree, so the initial context is present as a calculation. But you see what I'm getting at, so I'll take it. I hesitate to use the term "physicality"—it's still an abstract ordering in my mind, but this is inessential.

The Digital ZK ID doesn't specifically provide this type of context but does provide unique context. A date stamp is context and can be crucial to some future self. The zettel title is a powerful piece of context impossible to replicate with a folgezettel ID of '25.6.2.2'.

Example IDs - which provides more apparent context?
folgezettel - 25.6.2.2
digital - Folgezettel-Digital ID comparison 202110101626

Not fair @Will ! You would still have the title in addition to the Folgezettel ID. I didn't say only use Folgezettel IDs and scrap Zettel titles. Here's the steel man comparison:

Example IDs - which provides more apparent context?
folgezettel - Folgezettel-Digital ID comparison $(22.1/_3 25.6.2.2 /_5 1.1)$
folgezettel - 202110101626/1 Think Globally, Link Locally
digital - Folgezettel-Digital ID comparison 202110101626

However, the steel man is molten and melts into an unspecified, sputtering blob upon the realization that 1-the generalized Folgezettel IDs would have to be translated to an operating-system compatible filename format to use them in Markdown [[...]] links or [...](...) links, for that matter, (very doable, with other conventions); and 2- the renaming of files to match the IDs whenever they are created is too much to ask of anyone, in the absence of software support for this type of ID (doable, with enough effort, not a priority, much less a Zettelkasten emergency).

But if you use YAML, as you suggest, they can go in the YAML header (provided your software will search on them), or you can add them below the title somewhere. The point is to locate the notes in the tree you have in mind, and to augment the search.

They could be hashtags, once translated into something usable. (I have some regexes upthread where I attempted a translation, but without the hashtag.) I take that back: hashtags will clutter the keyword database.

Better: use file aliases. Obsidian supports them. I don't know if Zettlr does. I'm reluctant to switch to Obsidian. Absurdian!

I say all this not to convince you to cry, uncle. I do think there is something there here. You might be starting to hit on it.

I suppose I hadn't been clear. In any case, there is progress.

Don't cry, smile because we got to here.

As for crying, I wouldn't deny lachrymal replenishment to anyone who might drink my tears.

I'm testing these ideas by speaking out loud. "I don't know what I think until I write it down."^{1}

@ZettelDistraction said:
Not fair @Will ! You would still have the title in addition to the Folgezettel ID. I didn't say only use Folgezettel IDs and scrap Zettel titles. Here's the steel man comparison:

Example IDs - which provides more apparent context?
folgezettel - Folgezettel-Digital ID comparison $(22.1/_3 25.6.2.2 /_5 1.1)$
folgezettel - 202110101626/1 Think Globally, Link Locally
digital - Folgezettel-Digital ID comparison 202110101626

However, the steel man is molten and melts into an unspecified, sputtering blob upon the realization that 1-the generalized Folgezettel IDs would have to be translated to an operating-system compatible filename format to use them in Markdown [[...]] links or [...](...) links, for that matter, (very doable, with other conventions); and 2- the renaming of files to match the IDs whenever they are created is too much to ask of anyone, in the absence of software support for this type of ID (doable, with enough effort, not a priority, much less a Zettelkasten emergency).

Yes, now I see just how cheap a shot it was. I have to leave aside my confabulated ideas about folgezettel ID's. We are talking about them without handicapping them on physical notecards. If we limit our ideas about the IDs in question to just the numerical, $(22.1/_3 25.6.2.2 /_5 1.1)$, or 202110101626, we may make some headway. The title could be optional with either ID.

With a folgezettel ID, you get a scaffolding much like a cryptic outline or map of contextual placement of conceptual order. There is a supertanker amount of benefit seeing the contextual placement of an idea.

With the digital Zk (Time Based) ID, you get a different scaffolding that circles the mundane notion of contextual placement based on temporal relationships. "What where the ideas I thinking about last Thursday? How are they related?"

Comparing just the IDs, it becomes clear the folgezettel method is the genuine queen bee.

But if we extend our thinking and add structure notes, we find things shifting towards the drones in the hive. (I love the metaphor!) Structure notes should be used with either ID scheme. Making the context of a note/idea malleable, but the downside with folgezettel IDs is you lose the temporal context and can be misleading following the ID's scaffolding when a newer, more relevant set of relationships is outlined; elsewhere (in a structure note.)

I'm reluctant to switch to Obsidian. Absurdian!

Ha! Ha!

I suppose I hadn't been clear. In any case, there is progress.

I suppose I hadn't been clear. In any case, there is progress.

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

20211011124900 Proof that structure notes subsume Folgezettel IDs.

This note is to telegraph a proof that structure notes will reproduce the generalized Folgezettel outlines of outlines of outlines ..., based on the notation we developed. It has the virtue of telling us where to add structure notes, given an assignment of Folgezettel IDs. But first, a digression that led to this.

@Will said:
Structure notes should be used with either ID scheme.

They can be. I'm going to provide a procedure for deciding where to add structure notes. This was missing. The idea is to identify a structure note with the root of a subtree. It follows that structure notes subsume Folgezettel.

Given: a ZK where the notes have been assigned general Folgezettel IDs. We may assume that all Zettel IDs have the form $(Z/_n d)$. That is, without loss of generality, we assume that no Zettel has the ID $(d)$, where $(d)$ is a (positive) decimal.

If $(Z/_n d)$ is the ID of a note in the ZK, then create a structure note with structure note ID $(Z/_n)$. Note that prior to the addition of this note, $(Z/_n)$ is not the ID of any note in the ZK. This naming enables us to recognize structure notes by ID.

Within the structure note $(Z/_n)$, create a link [[...]] to every note in the ZK of the form $(Z/_n d)$, for each such positive decimal $(d)$.

This procedure gives the desired translation.

Making the context of a note/idea malleable, but the downside with folgezettel IDs is you lose the temporal context and can be misleading following the ID's scaffolding when a newer, more relevant set of relationships is outlined; elsewhere (in a structure note.)

Folgezettel can handle this case too, but at a cost. The general Folgezettel IDs can be interpreted as version numbers. More relevant relationships can be specified with newer Folgezettel IDs for commentary on related branches, at the possible cost of duplicating (or aliasing) some notes that may remain in the previous ordering. (Luhmann introduced hub notes for cases like this. There is no essential difference between hub notes and structure notes.)

The utility of structure notes is that they can preserve the initial subtree structure of the outlines below them. A new or revised structure note can add references to new notes along with references to pre-existing notes. With Folgezettel and without hub notes, it is necessary to either copy pre-existing notes under the new root of the new subtree, or else create a new note under the new subtree whose purpose is to refer the old note (effectively creating an alias). Wasteful either way. So structure notes provide an efficient versioning function.

The following claim has been frequently asserted.

Structure notes are the digital replacement for Folgezettel [note relationship].

How do we know this? Do we lose any combinatorial information? I wanted a proof. Now we have a translation proof, though the procedure retains older structure notes and creates new ones as relationships change. This would be the case with Luhmann's Zettelkasten.

Sketch of the converse

Generalized Folgezettel IDs subsume digital ZK with timestamp IDs, possibly with some loss of information.

Sketch of proof.
1. Given a digital ZK with timestamp IDs only, we may assume that the ZK is a connected graph of notes (the general case follows).
2. Construct a spanning tree of the ZK and assign Folgezettel IDs starting with the root. If there are several connected components, assign Folgezettel IDs to each tree of the forest of $(k)$ spanning trees to avoid ID collisions, say by numbering the $(k)$ roots with prefixes starting from $(1)$ through $(k,)$ as in: $$(1/_1 1,2/_1 1,\cdots, k/_1 1.)$$
3. If $(Z/_n d)$ is the ID of a node with $(k)$ subtrees, assign the subtrees the IDs
$$(Z/_n d /_1 1, \ldots, Z/_n d /_k 1)$$ in order. Other assignments are possible, such as the assignment of $(Z/_n d.1)$ to the first subtree, and
$$(Z/_n d /_1 1, \ldots, Z/_n d /_{k-1} 1)$$ to the remaining $(k-1)$ subtrees, if $(k\gt 1.)$
4. Systematically replace timestamp links within each note with the corresponding Folgezettel references. (We assume we record the timestamp ID to Folgezettel ID mapping, implicitly defined by this process, in order to carry out this step. In fact, it would be sufficient to construct such a mapping from timestamp IDs to Folgezettel IDs, extending at each step. This is how the proof should be rewritten. )

Good enough. It looks a bit like elementary (really elementary) model theory.

Concluding Unsympathetic Postscript

I still think Folgezettel IDs are useful, at least in the beginning, because they indicate where structure notes can be added. With Folgezettel IDs, you don't have to master the art of deciding where to add structure notes. In the absence of Folgezettel, you'll have to develop judgment where to add them.

Conversely, a digital ZK with timestamp IDs can be relabeled with Folgezettel IDs, though we didn't constrain the spanning trees and the numbering so that the lexicographic order associated with the Folgezettel numbering corresponds with the timestamp order within each spanning tree. There is no reason why the numbering we chose would correspond to the order in which notes were created, much less reflect their initial placement. Choosing spanning trees so that later timestamps lie below earlier timestamps looks like a standard combinatorial problem. There is still loss of information, even if the lexicographic ordering on the linearized Folgezettel IDs respects the timestamp order, since we can't say (without a reading each note, and even then there are no guarantees) whether a successor note is a continuation, an aside or the start of a new topic. Nevertheless, anything digital IDs can do, Folgezettel can do. To be continued...

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

20211011124900 Proof that structure notes subsume Folgezettel IDs.

Given: a ZK where the notes have been assigned general Folgezettel IDs. We may assume that all Zettel IDs have the form $(Z/_n d)$. That is, without loss of generality, we assume that no Zettel has the ID $(d)$, where $(d)$ is a (positive) decimal.

If $(Z/_n d)$ is the ID of a note in the ZK, then create a structure note with structure note ID $(Z/_n)$. Note that prior to the addition of this note, $(Z/_n)$ is not the ID of any note in the ZK. This naming enables us to recognize structure notes by ID.

Within the structure note $(Z/_n)$, create a link [[...]] to every note in the ZK of the form $(Z/_n d)$, for each such positive decimal $(d)$.

I missed your definitions in the haystack of maths. What is an example of the differences between $(Z/_n d)$ and $(Z/_n)$?

The utility of structure notes is that they can preserve the initial subtree structure of the outlines below them. A new or revised structure note can add references to new notes along with references to pre-existing notes. With Folgezettel and without hub notes, it is necessary to either copy pre-existing notes under the new root of the new subtree, or else create a new note under the new subtree whose purpose is to refer the old note (effectively creating an alias). Wasteful either way. So structure notes provide an efficient versioning function.

I prefer to view structure notes as providing a Darwinian evolutionary function. They evolve as my knowledge within the structure evolves. I feel no need to keep old thinking around for nostalgic reasons.

Until now, my main beef with folgezettel IDs is that I can't see how they scale past 10 or 20 notes as a guide to outlining the context. I know that somehow Luhmann managed to use folgezettel IDs for many thousands of notes. There must be something I'm not grasping. Tutor me!

I have an idea for retrospectively adding folgezettel IDs, but how might one do this prospectively with 24 hubs and 30 structure notes under them.

My 2348th zettel has an ID of 202110120749
Niklas Luhmann's 2348th zettel has an ID of 21/3a1p5c4fC1a^{1}
How can any sense be made of Luhmann's ID, mathematical or otherwise?

Concluding Unsympathetic Postscript

I still think Folgezettel IDs are useful, at least in the beginning, because they indicate where structure notes can be added. With Folgezettel IDs, you don't have to master the art of deciding where to add structure notes. What contextual information could an ID of 34.6.7,12 possibly

An upper-crustian (I almost wrote crustacean) read.

You've said elsewhere that you come from a family of artists and writers. Someone in your circle has talked about Joan Didion as a writer? Have you read this book by Joan Didion?

Concluding Unsympathetic Postscript [Take Two]

I still think Folgezettel IDs are useful, at least in the beginning, because they indicate where structure notes can be added. With Folgezettel IDs, you don't have to master the art of deciding where to add structure notes. In the absence of Folgezettel, you'll have to develop judgment where to add them.

Conversely, a digital ZK with timestamp IDs can be relabeled with Folgezettel IDs, though we didn't constrain the spanning trees and the numbering so that the lexicographic order associated with the Folgezettel numbering corresponds with the timestamp order within each spanning tree. There is no reason why the numbering we chose would correspond to the order in which notes were created, much less reflect their initial placement. Choosing spanning trees so that later timestamps lie below earlier timestamps looks like a standard combinatorial problem. There is still loss of information, even if the lexicographic ordering on the linearized Folgezettel IDs respects the timestamp order, since we can't say (without a reading each note, and even then there are no guarantees) whether a successor note is a continuation, an aside or the start of a new topic. Nevertheless, anything digital IDs can do, Folgezettel can do. To be continued...

Your discussion of 'relabeling' is timely and relevant to one of my near-term future projects. You pointed out problems wanting solutions. Preserving the "order in which notes were created" and " later timestamps lie below earlier timestamps" are things I hadn't considered yet.

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

Anything digital timestamp IDs can do, general Folgezettel IDs can do, and conversely. Because of the potential for loss of information, it's a good idea to decide on one scheme from the outset, (or do both at once, which is possible).

@Will said:
I missed your definitions in the haystack of maths. What is an example of the differences between $(Z/_n d)$ and $(Z/_n)$?

$(Z/_n d)$ is a Folgezettel ID, such as
$(2.1/_5 1.1.3)$. It's not a structure note. It's note 1.1.3 of the 5-th alternative branch under note 2.1.

To make a structure note for the notes under the 5-th branch of 2.1, create a new note with the ID $(Z/_n)$, which in this example would be $(2.1/_5 1.1.3 /_1 7)$ By construction it cannot be the ID of an existing note. Next, we add all the IDs of notes of the form $(Z/_n d)$, where d is a decimal. (We will get to longer prefixes later). This means we add $(2.1/_5 1)$, ..., $(2.1/_5 1.1.3)$ and others, but we would not add $(2.1/_5 1.1.3 /_1 7)$, for example. That would belong to a structure note with the ID $(2.1/_5 1.1.3 /_1 )$ which would then link to $(2.1/_5 1.1.3 /_1 7)$.

Now I know that $(2.1/_5 1 )$ was the first note of the 5-th alternative branch of 2.1. I could link the structure note $(2.1/_5 )$ to 2.1, for example, if I wanted. But the purpose is the build up structure notes in a tree that doesn't have them.

l

I prefer to view structure notes as providing a Darwinian evolutionary function. They evolve as my knowledge within the structure evolves. I feel no need to keep old thinking around for nostalgic reasons.

There is no requirement to keep older versions around. The point of this construction is show that ZK with structure notes will do everything that Folgezettel will do.

Until now, my main beef with folgezettel IDs is that I can't see how they scale past 10 or 20 notes as a guide to outlining the context. I know that somehow Luhmann managed to use folgezettel IDs for many thousands of notes. There must be something I'm not grasping. Tutor me!

I attempted to interpret one of Luhmann's IDs below. And I rewrote it in my notation.

I wrote down a system the could be described and used mechanically.
Luhmann's isn't so tractable.

I have an idea for retrospectively adding folgezettel IDs, but how might one do this prospectively with 24 hubs and 30 structure notes under them.

If hub is a note serving as an entry point, and the structure is a forest of trees...

To simplify the notation, write $(/)$ for $(/_1)$. Number the "root" notes of the 24 hubs with these IDs.
1/1, 2/1,... ,24/1.
Consider 1/1.
Case 1: the next note is a continuation of 1/1. The next note has ID 1/1.1 or 1/2.
[Aside: a note after 1/1.1 could have ID 1/1.1.1, 1/1.2 or 1/2. Likewise a note after 1/2 could be 1/3 or 1/2.1. The choice is of lexicographic successors.]

Case 2: the next note comments on 1/1 but is not a continuation. Here you have a choice. The there could be several such comments. Suppose this is the third such comment. Assign it the ID $(1/1/_3 1)$.
This is where outlines of outlines come in.

Case 3. The next note belongs to another hub. Do nothing until you run out of notes for this hub.

Case 4 structure note. I wouldn't treat these differently from any other note in this case. Either you think of it as a continuation of (some note under) 1/1, or an alternative.

My 2348th zettel has an ID of 202110120749
Niklas Luhmann's 2348th zettel has an ID of 21/3a1p5c4fC1a[^1]
How can any sense be made of Luhmann's ID, mathematical or otherwise?

21/ looks like category 21.
21/3 is the third note of category 21, starting with 21/1, continuing with 21/2 and continuing further to 21/3.
$(21/_1 3)$ in my notation
21/3a is the first comment on 21/3
$(21/_1 3/_1 1)$ in my notation
21/3a1 is a comment on 21/3a1
$(21/_1 3/_1 1/_1 1 )$ in my notation
21/3a1p is a continuation of a sequence of notes starting with 21/3a1a, which was a comment on 21/3a1. 21/3a1a in my notation is
$(21/_1 3/_1 1/_1 1/_1 1 )$, and the p-th item in the sequence 21/3a1p in my notation is
$(21/_1 3/_1 1/_1 1/_1 16 )$

21/3a1p5 is the 5th item in a sequence, starting with 21/3a1p1, which was itself a comment on 21/3a1p. In my notation, 21/3a1p5 is
$(21/_1 3/_1 1/_1 1/_1 16 /_1 5)$
Luhmann allows one comment, it looks like, so far anyway. My notation allows for any number of comments on a given note. Perhaps there is an exception, but let's write $(/)$ for $(/_1)$ to simplify notation.

21/3a1p5 is then
$(21/ 3/1/ 1/ 16 / 5)$

21/3a1p5c4f is
$(21/ 3/1/ 1/ 16 / 5 /3 /4 /6 )$
We can interpret this as the 6th sequential note of a comment on the 4th note of a comment on the 3rd note of a comment on the 5th note of a comment on the 16th note of a comment on a comment on the 3rd note of the category or section numbered 21.

We could continue, but suppose for the sake of argument that whenever Luhmann wanted to comment on more than one aspect of a note, say 21/3a1p5c4f
he used a letter followed by another letter, followed by a number

21/3a1p5c4fC1a would then be
$(21/ 3/1/ 1/ 16 / 5 /3 /4 /6/_3 1 /1)$

This might not be right, but it says that this is the first comment on a the third alternative comment on the 6th note of a comment ...

But maybe not. Maybe capital C is just 3, so we get

$(21/ 3/1/ 1/ 16 / 5 /3 /4 /6/3 /1 /1)$
For the whole thing.

His notation is more suited to at most one comment on an aspect of a single card.

An upper-crustian (I almost wrote crustacean) read.

I had deleted this from an earlier version and thought better of it. Now I am going to pay.

You've said elsewhere that you come from a family of artists and writers.

Yes.

Someone in your circle has talked about Joan Didion as a writer?

My mother liked her work--I think.

Have you read this book by Joan Didion?

A while back. I seem to recall spur of the moment flights to Paris, entertaining contingency plans that would only have occured to the wealthy and well connected,--but perhaps this or some of it was magical thinking. Now I am going to have to reread about the death of a famous writer's husband. I like reading medical reports, so there's something to look forward to.

Your discussion of 'relabeling' is timely and relevant to one of my near-term future projects. You pointed out problems wanting solutions. Preserving the "order in which notes were created" and " later timestamps lie below earlier timestamps" are things I hadn't considered yet.

I have to think this through more carefully. 笑。 If there are global constraints, such as "there must be 30 structure notes," then this might not be so feasible. I may have an integer linear programming problem to solve.

Luhmann worked by finding where a new note could fit in, and numbered accordingly. Presumably his "structure notes" were numbered like any other note...

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

@Will
The notation $(Z/_n d)$ is supposed to handle notes arranged in outline format, with some notes commenting on on aspects of others, as in Luhmann, although in my case the commenting note could be the start of another outline. Luhmann's system is a special case. It may be that Luhmann did have such outlines in mind, but this isn't obvious to me with 21/3a1p5c4fC1a. The interpretation 21/3/1/1/16/5/3/4/6/3/1/1 seems more accurate than 21/3.1.1.16.5.3.4.6.1.1.

While I'm strapped for time, I started rereading Joan Didion's book. I ascribe to untreated severe sleep apnea whatever impression I had about it years ago. I have somewhat more patience (for reading through detailed personal timelines) than I had then. Now I'm more interested in her technique.

Another point: 21/3a1p5c4fC1a is an address of a card. Each card was added one at a time somewhere in the stack, depending on its relation to the preceding card. The address 21/3a1p5c4fC1a is the first comment on an aspect of 21/3a1p5c4fC1. If you had to assign addresses to 67,000 cards with constraints on where they could go, you could be facing a combinatorial explosion. The addresses weren't thought of in advance. They grew over time. Luhmann had a keyword database to locate addresses of relevant cards to supplement the linking system. The unreadability of 21/3a1p5c4fC1 is probably an advantage for someone who wants to be surprised. It's likely that not every card that could be assigned a keyword had that keyword. It's more likely to have been linked to a card that had the keyword. That would make it necessary to follow links--and who knows where that might lead.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

@ZettelDistraction said:
To simplify the notation, write $(/)$ for $(/_1)$. Number the "root" notes of the 24 hubs with these IDs.
1/1, 2/1,... ,24/1.
Consider 1/1.
Case 1: the next note is a continuation of 1/1. The next note has ID 1/1.1 or 1/2.
[Aside: a note after 1/1.1 could have ID 1/1.1.1, 1/1.2, or 1/2. Likewise, a note after 1/2 could be 1/3 or 1/2.1. The choice is of lexicographic successors.]

Case 2: the next note comments on 1/1 but is not a continuation. Here you have a choice. There could be several such comments. Suppose this is the third such comment. Assign it the ID $(1/1/_3 1)$.
This is where outlines of outlines come in.

Case 3. The next note belongs to another hub. Do nothing until you run out of notes for this hub.

Case 4 structure note. I wouldn't treat these differently from any other note in this case. Either you think of it as a continuation of (some note under) 1/1, or an alternative.

I'm still struggling to see how this scales in the scenario you laid out.

Thought Experiment
You land in a perfectly folgezettel'd zettelkasten about to create note number 2348.
You have an idea spurred by an article in the Lion's Roar about the role of scatological humor in Haiku, and your memory is unusually sharp today. You remember that Hub 21 is your Haiku Hub. '
Searching on IDs that start with 21, you'd be presented with a list of 40 + notes. (I get 40 by adding the sequence of numbers in the above ID, but the number is likely higher as some branches would have more notes than others. I think 40 would be a minimum.)
My question is, how would we determine the ID for the note describing the historical use of scatological humor in Haiku? Even given that the list would include titles and IDs. Would it be 21/3a1p5c4fC1a1 or 21/2b2r6D3? How much time should we spend trying to figure this out? What is the benefit of spending the necessary time to get this right? How bad would a mistake in note placement be?

Luhmann worked by finding where a new note could fit in and numbered accordingly. Presumably, his "structure notes" were numbered like any other note...

Confusion abounds. At the start of your prior post (quoted here), you say, "Number the "root" notes of the 24 hubs with these IDs. 1/1, 2/1,... ,24/1." Now you say, "Presumably, his "structure notes" were numbered like any other note."

The only benefit I can see is that this process forces you to spend time at note creation searching for a place to land the ID, so the sequence makes sense. Slowing down and thinking about how your note's idea integrates is a win. But, this is why I fail to see how this scales past 10-20 notes. The task of finding the prior ID from which to continue the thread gets more and more demanding as the zettelkasten builds. I envision a time when you'd have to spend herds-more-time finding the ID for a note than the time since the Big Bang until the Singularity.

Something about this process I don't understand: I can't imagine doing this without the help of a computer. Luhmann must have accomplished this using the spatial environment for clues. Maybe he would think hub 21 - Haiku was in drawer 16 about the middle. And maybe his keyword index would tell him that Haiku Humor started at 21/3a1p5, and his search would be limited to that 4 or 5 inches of notes in the middle of drawer 16, eventually seeing that this note follows 21/3a1p5c4fC1a with an ID of 21/3a1p5c4fC1a1. This seems unnecessary in the computer age when you could allow the computer to assign a unique ID faster than it takes light to cross the street and be done with it.

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon 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:
The only benefit I can see is that this process forces you to spend time at note creation searching for a place to land the ID, so the sequence makes sense.

Yes. That's exactly what Luhmann did.

Slowing down and thinking about how your note's idea integrates is a win. But, this is why I fail to see how this scales past 10-20 notes. The task of finding the prior ID from which to continue the thread gets more and more demanding as the zettelkasten builds.

That's true too. Luhmann spent a lot of time with his Zettelkasten maintaining it this way. There may be several places where a note might be linked to a related note.

I envision a time when you'd have to spend herds-more-time finding the ID for a note than the time since the Big Bang until the Singularity.

True also unless you had another means of searching through the notes. Luhmann had a keyword index. We have text searches, regular expressions, etc. Of course an astronomically large collection created by a genuine galaxy brain would begin to slow down over time...

Something about this process I don't understand: I can't imagine doing this without the help of a computer. Luhmann must have accomplished this using the spatial environment for clues. Maybe he would think hub 21 - Haiku was in drawer 16 about the middle. And maybe his keyword index would tell him that Haiku Humor started at 21/3a1p5, and his search would be limited to that 4 or 5 inches of notes in the middle of drawer 16, eventually seeing that this note follows 21/3a1p5c4fC1a with an ID of 21/3a1p5c4fC1a1. This seems unnecessary in the computer age when you could allow the computer to assign a unique ID faster than it takes light to cross the street and be done with it.

Let's work with a digital ZK with time stamps.

You can assign a time stamp ID $(T_1)$ to the new note, which we will give the alias 21/3a1p5c4fC1a1 so that we are referring to the same notes as in your example. In the digital ZK it has ID $(T_1)$, but we can refer to it outside of the ZK by this nickname. Also, outside the ZK we refer the note with timestamp ID $(T_0)$ as 21/3a1p5c4fC1a. The ZK only sees $(T_0)$ and doesn't recognize the alias.

Now suppose that the following assertions are true.

We locate the note with the alias 21/3a1p5c4fC1a, which has a timestamp ID $(T_0)$. Maybe we used keyword searches or something else. There could be more than one match.

We now have $(T_0)$, and we know the relationship it bears to $(T_1)$.

We link to $(T_0)$ from $(T_1)$

In $(T_1)$ we indicate the relationship $(T_1)$ bears to $(T_0)$: either $(T_1)$ is a continuation of $(T_0)$, or it comments on an aspect of $(T_0)$

If all of these assertions hold after we add $(T_1)$, then we have the same information with time stamp IDs that the ordinary Folgezttel IDs would have given us. (My IDs will give you more than one comment on a note, but that's not at issue.)

Optional. If we want the same information that scanning through the physical ZK would have given us when $(T_1)$ was added, we could add 5 and 6 below. However, as more cards are added there could be many cards separating 21/3a1p5c4fC1a from 21/3a1p5c4fC1a1. These steps are optional, because we are interested in preserving the information that the Folgezttel IDs alone would have given us. The longer ID 21/3a1p5c4fC1a1 informs us about the shorter ID 21/3a1p5c4fC1a, but the shorter ID just gives us the prefix of the IDs of related notes that may end up in remote precincts of the physical ZK, assuming that the series continues. Aha: and that is one reason to use structure notes.

We link to $(T_1)$ from $(T_0)$.

In $(T_0)$ we indicate the relationship $(T_0)$ bears to $(T_1)$: $(T_1)$ continues $(T_0)$, or it comments on an aspect of $(T_0)$.

So what?

If we don't do at least 1-4, we will end up with something else. Maybe better, about the same, or worse. But we need an argument to say that whatever we do come up with will yield at least as many combinatorial possibilities for "surprise" (the same collection of paths are available, or a close enough collection) as a ZK with Folgezttel IDs, or that it will meet certain design constraints, e.g., good enough for Luhmann (I am being vague).

Or we can say we don't care, it's not interesting etc.

If we just assign a time stamp ID to $(T_1)$ and leave it at that, well, you'll have my ZK, more or less.

Confusion abounds all around

I might not need this argument, except we wanted to build in structure notes

Confusion abounds. At the start of your prior post (quoted here), you say, "Number the "root" notes of the 24 hubs with these IDs. 1/1, 2/1,... ,24/1." Now you say, "Presumably, his "structure notes" were numbered like any other note."

What do you mean by a "hub"? I assumed you meant the top of a tree-like structure--that the hubs were the roots. I should not have referred to "root notes of hubs". Bad contradictory phrasing, given the assumptions. (This is why I leave my old notes in my ZK and create newer revisions rather that ditch them: to record and identify errors that I want to avoid in the future, such as careless formulations, overlooking cases, etc. I want to see my mistakes.) Maybe hub doesn't mean what I thought it means. If "hub" means "structure note," what would the constraint of 24 hubs and 30 structure notes mean?

The original construction was meant to solve a different problem, which is how to convince a skeptic that you could replace a ZK with Folgezettel with one that had timestamp IDs and structure notes, but not lose any paths through the ZK.

Unfortunately it was a sketch.

The starting position is a ZK with Folgezettel IDs set up in advance, but there are no structure notes. How would you add them? There was a procedure. The new notes have to be linked somewhere--there is a choice of nodes to link to them. Once they are added, replace all of the Folgezettel ids (and the new structure note ids) with timestamps, so that all the original notes have the same interconnections as before, only timestamps--the Folgezettel are erased. You can do this so that the tree ordering is linearized by timestamp order.
(I need the preceding arguments, which I should combine with two minor typos corrected.) The result is a digital ZK with at least as many paths as the original. Now I have to check this...

But the other direction is harder.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

One of the key takeaways from this discussion is that folgezettel ZK puts structural order in the ID of the note, and digital ZK simular structural order is placed in a separate structure note. A note can be part of many structural orders. In otherwords, have multiple folgezettel paths or IDs.

You can assign a time stamp ID $(T_1)$ to the new note, which we will give the alias 21/3a1p5c4fC1a1 so that we are referring to the same notes as in your example. In the digital ZK it has ID $(T_1)$, but we can refer to it outside of the ZK by this nickname.

Now suppose that the following assertions are true.

We locate the note with the alias 21/3a1p5c4fC1a, which has a timestamp ID $(T_0)$. Maybe we used keyword searches or something else. There could be more than one match.

We now have $(T_0)$, and we know the relationship it bears to $(T_1)$.

We link to $(T_1)$ from $(T_0)$ to indicate this.

In $(T_0)$ we indicate the relationship $(T_0)$ bears to $(T_1)$: $(T_1)$ continues T0, or it comments on an aspect of $(T_0)$.

We link to $(T_0)$ from $(T_1)$

In $(T_1)$ we indicate the relationship $(T_1)$ bears to $(T_0)$: either $(T_1)$ is a continuation of $(T_0)$, or it comments on an aspect of $(T_0)$.

The relationship between $(T_0)$ and $(T_1)$, as you say, might be a continuation or a comment on some aspect. $(T_1)$ could also be questioning, a refutation, a reference, or a quote somehow related to $(T_0)$. We have described linking irrespective of the $(T)$ or $(Z)$

If all of these assertions hold after we add $(T_1)$, then we have the same information with time stamps that the ordinary Folgezttel IDs would have given us. (My IDs will give you more than one comment on a note, but that's not at issue.)

It strikes me now that folgezettel IDs embed the structural order of the note in the ID, whereas, in a Digital ZK, the structural order is spelled out in a structure note. The Digital ZK is human-readable, and there is no space limitation to describe and annotate the structural order.

The note ID is of fractional importance. It is the interstitial links within the note that are important. They are the nodes showing relationships. They are what gives the ZK structure and mass.

What do you mean by a "hub"? I assumed you meant the top of a tree-like structure--that the hubs were the roots. I should not have referred to "root notes of hubs". Bad contradictory phrasing, given the assumptions. ... Maybe hub doesn't mean what I thought it means. If "hub" means "structure note," what would the constraint of 24 hubs and 30 structure notes mean?

Hub, in my view, is one of the synonyms for structure notes. It depends on what flavor of metaphor you like. Wheel, spoke, hub or first level, second level, structure. There are others. In my case, I have 24 primary hubs (I call these Garden Notes.) centered around key interset and areas of study. The 30 secondary structure notes have evolved naturally over time, and many of these are offshoots from the primary hubs when they get big and unwieldy.

The starting position is a ZK with Folgezettel IDs set up in advance, but there are no structure notes. How would you add them? There was a procedure. The new notes have to be linked somewhere--there is a choice of nodes to link to them.

Structure notes are added after a critical mass of notes. How many are we talking about? Let's focus on the ones that already have folgezettel IDs. If the notes contain tags, they might help with the step of starting a structure note. Ideally, a structure note is an "Annotated" table of contents and not a simple index. It might start as a simple index but should evolve by adding annotations. This doesn't have to be done in one go. I imagine in your case you'd have a Math hub (a top-level structure note) with maybe some second-level structure notes on, say, Catagory Theory, Math Creativity, such. The key to this is not to force it, to let these emerge naturally. At first copy to IDs and Titles to the page like:

- • ZK Zettelkasten [[202003231430]]
- G-Thinking Skills [[201812271440]]
- B-Designing creative workflows [[201901281916]]
- Effective Procrastination [[202102280816]]
- Attention And Distraction Are Inversely Related [[202103270623]]
- Inveterate backslider [[201901031421]]

This isn't ideal, but it is the first step. Then over time, as notes and structure notes are encountered, add notations and annotate them. It is always a moving target. I add to and modify annotation without hesitation. Here is a Stoic structure note sample where some of the links have a short annotation and have been broadly grouped. This work continues.

# • Stoic Philosophy
›[[202010110616]]
10-11-2020 06:16 AM
- #garden #stoicism
"The wind blows, the people form beliefs, the river flows, and in the end, the world swallows it all." [[201901021316]]
**Old age is no excuse for coasting**
Never Stop Trying To Get Better
- evernote:///view/597091/s5/26f9fdca-4c96-4625-a3cd-8f21fd1082b8/26f9fdca-4c96-4625-a3cd-8f21fd1082b8/
## View from above.
- Separate beliefs from reality [[201904040921]]
## Literature
• Meditations on Self-Discipline [[201901021303]]
-Written in the style of Meditations by Marcus Aurelius
## Opinions
- Don't Care About Anyone’s Opinion [[202006060707]]
- Personal Superpower
- Stop Caring What Other People Think [[202011190711]]
- Personal Superpower
- Be true to the principled one [[201903190550]]
- Don't broadcast [[201912120932]]
- Talking ape with shoes [[201901040626]]
- Don't be sheep [[201901031449]]
- Think the same thought [[201901031203]]
## Principles
- Stoic Principles [[202001121118]]
- Signs of Stoic Progress [[202001021636]]
- Life is ephemeral [[201901070542]]
- Embrace mortality [[201901021304]]
- Your Choices Make You [[202012161026]]
- Do you hit the snooze button, or get up early?

Once they are added, replace all of the Folgezettel ids (and the new structure note ids) with timestamps, so that all the original notes have the same interconnections as before, only timestamps--the Folgezettel are erased. You can do this so that the tree ordering is linearized by timestamp order.

I'm not sure you'd want to scrap the folgezettel IDs.

But the other direction is harder.

Maybe, we'll see.

Will Simpson
I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time. kestrelcreek.com

I have some pressing obligations and deadlines, have little time now and will be off the air until the weekend or possibly after.

We're at cross purposes in some of this. I had asked what you meant by hubs and structure notes. Thank you for setting me straight. Then, without warning I launched into a sketch of an argument against a Folgezettel Fundamentalist. It was a reprise--I should have said so. I also tend to edit after the fact. I appreciate your comments on this, because they were very helpful.

When I got to the part of the sketch where I said, "now replace everything with timestamps," the idea was to demonstrate to the Folgezettel Fundamentalist (or the person isolated from the rest of society by the Folgezettel Fundamentalist proselytizer, or someone on the fence), that timestamps can do all of the linking, with some possible loss of information. And there is a "principled" way of adding in structure notes to recover the lost information— or you can annotate the links to recover the information lost to erasure. Not that I want to erase all my Folgezettel (I don't have any yet!).

However, Folgezettel come with a "Think globally, link locally" procedure. Find the place where they fit and assign (or link with indications--that doesn't matter). The Folgezettel indicate the place where a new note either continues a preceding note or comments (questions, whatever) on it.
They are designed with the purpose of retracing the thoughts one had originally. Even sadly mistaken, radically misconceived thought, or thoughts that one would prefer to forget.

That's all that the Folgezettel Fundamentalist has, in the end: a certain "observance" and an audit trail. There are different ways of doing this, of course.

There are Luhmann's IDs and the more general IDs I described. There is nothing stopping anyone from creating indexes etc with Folgezettel. My IDs were for my own purposes: they can be used to create trees with any finite number of branches in outline form, with any number of versions of, comments on, questions about (translating into vulgar academese "interrogations of"), discursive digressions
on any section, subsection, subsubsection, and so on of any section, subsection, subsubsection etc arising from a digression, etc. They are general tree or outline coordinates. With linking, they can implement version control.

I wanted to combine this way of thinking with digital ZK to audit the progression. No Spencerian Social Darwinism of Zettels: I'm keeping the petrified wood, the mammoths in permafrost, insects in amber, records of mass Zettel extinctions, and the fossils.

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

Again, I really enjoy the way you're taking a sledgehammer to crack a nut by employing mathematical formalizations to demonstrate in the end that you can map Folgezettel to whatever-you-want-IDs, including timestamps and "just use the title"-IDs I think, and not lose any information in the process. That's easy to assert, and I've tried to demonstrate this in the past, but not with such rigor

When cleaning things up, I wonder if a proper introduction to your maths would need to carefully explain that, even though Luhmann's IDs were e.g. of the form 21/2a5b19f, to make the expressions in the proof simpler, you're substituting characters with their position in the alphabet and consistently separate numbers from another with slashes. Or something like that. I'm afraid otherwise the generalization might go over readers's heads -- "aaaacktshually Luhmann's IDs were different, I disproved you!" -- and that would be a shame.

I also wonder what to do with the point of two-way translation being a problem: you can't get back from timestamp-based IDs to Luhmann IDs easily. That could kind of sound like timestamps are inferior in some reason ("ha-ha! You can't translate back, so they lack some information, I disproved you!!!1") while the opposite is true: you can have a multitude of equally footed structure notes that provide different views into your Zettelkasten. And since no order from these structures is superior to another in general (though one can be more useful than another for each individual, practical task), there's no favored order to use as a template to reconstruct nested IDs.

Now there's two camps that'd react differently:

The "IDs are hierarchical" camp: they could state that the important hierarchy of notes can be mapped onto e.g. timestamp IDs, but not back, because there's no preferred order, thus no actual hierarchy, thus a loss of information. That's bad.

The "IDs are not hierarchical" camp, which I find hard to grasp and summarize, so I might get them wrong: the initial Luhmann ID order didn't matter before, so it doesn't matter how you reassemble. There's no deterministic inverse operation? No mapping-back that guarantees the result to equal the original input? Doesn't matter. Just use any (?) combination of structure notes as input to have some kind of sensible (for humans) proximity in the Luhmann IDs.

@ctietze Thank you for these thoughtful comments--I will address them in a later revision after combining 20211004235454 Formalization and Generalization of Niklas Luhmann's Folgezettel IDs 1 and 20211006000400 Formalization ... 2 into a single document with a better title, "Folgezettel Formalized," and correct two errors from Formaliziation 2:
1. The $(w)$ in "Case 1. The case of $(w\in\mathcal{D}^+)$" should be a $(y)$.
2. The last line in the proof of surjectivity of $(L)$ should be $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}d =
L(x)\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\,\text{zeros}}d = L\left(x\left.\right|_n d\right).
)$$

Then there should be a note with better diagrams that could be dovetailed with the text.

@ctietze said:
Again, I really enjoy the way you're taking a sledgehammer to crack a nut...

Forthcoming sledgehammer iterations will add video-enabled remote sensing, redundant software-defined radios for spread-spectrum EHF telemetry, a 16-unit 64-core Raspberry pi 4 model B computational cluster, gyroscopic stabilization, wings and a Vespa mandarinia stinger.

When cleaning things up, I wonder if a proper introduction to your maths would need to carefully explain that, even though Luhmann's IDs were e.g. of the form 21/2a5b19f, to make the expressions in the proof simpler, you're substituting characters with their position in the alphabet and consistently separate numbers from another with slashes. ... I'm afraid otherwise the generalization might go over readers's heads -- "aaaacktshually Luhmann's IDs were different, I disproved you!" -- and that would be a shame.

It's indicated in "Formalization 2", but you're right, it should be spelled out. Unless the translation is explicit I have no more defense against "aaaacktshually" than I have against "OK, boomer." During the pandemic I gave the question of responses to "OK, boomer" considerable thought, and concluded that there is no response.

I also wonder what to do with the point of two-way translation being a problem: you can't get back from timestamp-based IDs to Luhmann IDs easily. That could kind of sound like timestamps are inferior in some reason ("ha-ha! You can't translate back, so they lack some information, I disproved you!!!1") while the opposite is true: you can have a multitude of equally footed structure notes that provide different views into your Zettelkasten.

I'm going to spend some time thinking through translations. Carefully.

Now there's two camps that'd react differently:

The "IDs are hierarchical" camp: they could state that the important hierarchy of notes can be mapped onto e.g. timestamp IDs, but not back, because there's no preferred order, thus no actual hierarchy, thus a loss of information. That's bad.

This is premature, but I think the general case is computationally hard. It's possible to find spanning trees in connected components and create IDs based on this, but the spanning trees have no relation to the timestamp ordering without additional constraints, and the timestamp ordering might be misleading.

The "IDs are not hierarchical" camp, which I find hard to grasp and summarize, so I might get them wrong: the initial Luhmann ID order didn't matter before, so it doesn't matter how you reassemble. There's no deterministic inverse operation? No mapping-back that guarantees the result to equal the original input? Doesn't matter. Just use any (?) combination of structure notes as input to have some kind of sensible (for humans) proximity in the Luhmann IDs.

Mapping from IDs to timestamps to IDs is doable, since the order in which timestamps are assigned can be controlled, it's possible to compute the neighborhoods of nodes and their degrees, and the timestamps are strictly increasing. Depending on the ID to timestamp direction it "should" be possible to reconstruct what you had. But once new notes are added, who knows.

The initial Luhmann Folgezettel IDs do matter--they were designed for a researcher (or a writer) who proceeds by reading (solving problems, writing) whatever they feel like reading (or solving, or writing)—ideally, just beyond their level; taking notes (writing solutions); and finding one of the several places in the ZK where the note might be patched in; and assigning an ID (or linking it). That's how the IDs could grow over time (if you use Folgezettel IDs). The intention isn't to interpret a long ID string, although one can try.^{1} The intention is "Think Globally, Link Locally," i.e., to support a certain bottom-up way of working on one or more sources (or targets).^{2} This ZK design is intended to keep track: to resume adding to or commenting on a sequence of notes; to reconstruct the original train of thought captured in a sequence;^{3} or to go beyond a sequence by following links and references leading away from it.^{4}Piotr Wozniak's incremental reading in SuperMemo comes to mind, with the exception that trains of thought in SuperMemo run on schedule.

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. ↩︎

I like to read several books and articles at once. ↩︎

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

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. ↩︎

Post edited by ZettelDistraction on

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

## Comments

## Mathematical definition of folgezettel and operatic lyricism 202110010741

I'm still wondering what is so fascinating about your mathematical dissection of the folgezettel. This morning it struck me musically. The symbology is the music, and your vocabulary being like operatic lyrics. First, you take one tack, then you quickly move to another. Sometimes quick and suddenly lingering on a detail.

I bounce in and out of your stream of proof. It is fast-moving, and I'm floating along up to my neck, getting glimpses of insight now and then as my head bobs along.

ps. I'm stealing your brilliant idea of posting some #forum-posts as zettel. I can draft them in The Archive, which is a more comfortable writing and thinking environment.

#math #beautiful-language

Will Simpson

I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time.

kestrelcreek.com

## 202110224430 Reply to 202110010741@Will

Thank you. I too have borrowed your discovery that keywords work in the forum. With posts in Zettelform, an organic development, the forum could itself become a collaborative digital Zettelkasten—if the hosts don't object.

Users would see different Zettel formats in action, and their possibilities. Over time it could become what Niklas Luhmann called a "communication partner." This phrase seems epistemically loaded: taking Luhmann at his word, it suggests that Luhmann and the Zettelkasten achieved common knowledge. Luhmann knew that the slip box knew what Luhmann added to it, and the slip box knew that Luhmann knew that the slip box knew, and so on. Perhaps the online digital Zettelkasten could achieve what virtual public spaces so far cannot do, which is to foster common knowledge, as opposed to distributed information.

Borrowing an idea from identity management, I referenced your post as "202110010741@Will," which will suffice here, though not in a global distributed Zettelkasten...

P.S. I owe @Sascha a reply.

#zettelkasten #scifi #commonknowledge

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

This must be the funniest joke in the History of Zettelkasten Science. What a strange loop you are.

I am a Zettler

## 20211002151545 Longueurs in reply to @Sascha

Do you suppose that humans have "agency"? They think and talk as if they do [Spinoza,

Ethics, Book 2, PROP. 35, NOTE, Spinoza.Ethics. PROP. 48, Spinoza.Ethics. Book 3, PROP 2, NOTE].In any case,

TL;DR: for 1. emulate Luhmann's system in a digital ZK; augment Zettel format to help track links; write update rules; write down precise mathematical definition of Folgezettel, partly to illustrate the difference between verbal theories and mathematics; write up a specific implementation "Digital Zettelkasten Step-by-Step." Need to review Schmidt.

For 2, still thinking...I am probably misreading Luhmann, or I haven't read enough.

A. [skip to B] Ultimately, I don't know what I am aiming at. I am driven to aim at

something. [I say driven instead of the word passion, a word employers have come to expect to hear from employees describing their attitude toward their employment, whatever their working conditions.] God willing, thatsomethingI happen to be aiming for does exist and is worth aiming for; or it doesn't exist, but I have neverthlesss aimed close enough in the direction of something that does exist and that is worth aiming for; or it doesn't exist, there is nothing close enough that is worth aiming for, but the result of pursuing something nonexistent was worth the effort despite my abysmal ignorance.A/1. Whatever reasons I might give for aiming at something might or might not be acceptable as reasons. Philosophers have never agreed on what counts as a reason in the 2500 year history of philosophy.

A/2. Then again, I could be ineffective in ways that a more experienced practitioner could identify.

B. But originally, I wanted to develop a checklist for working with Zettelkasten.

B/1. And I wanted to enhance the Zettel format I use, if necessary.

C. Niklas Luhmann was a sociologist who thought in terms of systems.

D. He developed and refined his Zettelkasten as a system for research and for thinking generally.

E. I wanted to see if Luhmann's system could inform my efforts, using a digital ZK.

E/1. I myself am interested in some of the applied mathematics of sociology developed since at least the middle of the 20th century, as this applies to some sociological problems of importance.

F. Luhmann's system relied on a numbering scheme that others have called Folgezettel.

G. He explains that his system was designed to introduce an element of surprise into his thinking, a middle ground between predictable order and utter chaos.

H. Since he regarded his ZK as a "communication partner," he had to surprise it. And it had to surprise him.

J. There are various mathematical characterizations of random processes that fall between predictable order and chaos.

^{[CITATION NEEDED]}There are also measures of surprisal, Kullback–Leibler divergence $(\displaystyle D_{\text{KL}})$ (also called relative entropy). But we are far from measuring this in a Zettelkasten.K. Luhmann claimed that his note numbering scheme, together with internal linkings of notes, worked to introduce a useful, exploitable type of randomness that would facilitate further thinking and writing. Enough to earn his keep as an academic—more than enough, even today

L. Encoded in the numbering scheme is the decision to introduce notes that continue a line of thought, that introduce a side-thought, or that start a new thought altogether.

M. Verbal descriptions of processes are likely to suffer from ambiguity. In [Gobet, F., Lane, P. C., & Lloyd-Kelly, M. (2015). Chunks, Schemata, and Retrieval Structures: Past and Current Computational Models. Frontiers in psychology, 6, 1785] Gobet et al write, "...not enough constraints are provided by verbal theories, and thus too much freedom is left in the way they can be interpreted."

N. I thought that I needed a mathematical description of Folgezttel to overcome the limitations of verbal theories.

O. One might object that Folgezettel are so simple that a verbal description is enough to avoid ambiguity.

P. But the Internet shows considerable back and forth over the significance of Folgezettel. And a verbal description isn't a programmable formula. The main issue is whether Luhmann's design did provide the middle ground between predictable order and chaos.

Q. A mathematical description will provide something to reason with.

R. Links can emulate Folgezettel--that is, links can express the same adjacency relations between notes as Folgezettel, but they do not record the decision that was made at the time a note was initially added to a ZK. This has to be added separately--by adding to the Zettel format, for example.

R/1 Links can emulate total disorganization or predictable order. They are unconstrained without a process.

S. Maybe the random graphs of probabilistic combinatorics are enough. Again there is the question of measurement and procedure.

T. Structure notes have been proposed as an alternative to Folgezettel. The question then is when to add them.

U. We are still far from characterizing the kinds of graphs and procedures likely to facilitate "productive" effort. This is assuming there is a combination of them that have some productive effect. The null hypothesis is that no graph configuration or procedure for extending a graph has any measurable effect.

V. But we have nothing to calculate with, nothing to measure the "semantic distance" between notes (I'll get back to you--maybe work of Bob Coecke and others in the UK is relevant), as this played a role in Luhmann's process. Only verbal theories.

V/1. Not that working systems require a theory. Most systems are under-explained and continue to function without the system realizing that no one understands how it works. The (macro-)economy is one such system.

W. Sorry for being such a wet blanket.

X. Still, I wanted a description for myself, with a comprehensible system for producing a ZK along Luhmann's lines, one that could readily be translated into a digital ZK.

Y. The linearization is an easily programmable map--it's a substitution.

Z.And I wanted to push the notion of Folgezettel somewhat, and for that, I needed mathematical definitions.

GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

## 20211002212456 Counterexample!

Comrades, fellow Zettlers, Zettelscenti, tovarishch Commissars: you missed an opportunity to draw attention to a spectacular public screw up. The generalized Folgezettel numbers survive, there are linearizations, but $(L)$ is too good to be true.

Counterexample. The map $(L)$ is surjective but not injective.$$(

L\left(1\mathbf{.}2\mathbf{.}3\right)=L\left(1\left.\right|_2 3\right)=

1\mathbf{.}2\mathbf{.}3.

)$$

There are many $(L)$-preimages of $(a.b.c.d)$, for example,

$$(

a.b.c.d, a|_b c.d, a.b |_c d

)$$ are preimages.

The map $(L)$ should be redefined (on finite subsets) to be order-preserving and injective. A question of shifting images out of the way. It's always possible to prepend a large enough integer to the value...but unless this is done carefully, it won't reproduce the "inner expansion" of Luhmann's system.

No, you're mistaken. Time for more "story editing."

Story editing and deflection. People make errors like this all the time, with the exception of historical figures like Gauss, Newton, Noether, John von Neumann, Erdös, Grothendieck, Gödel, and of course, Richard Stanley. Besides, a linearization isn't needed in the digital case.Rejoinder of the Default Mode Network. This should have been obvious to anyone, even at 3 AM in the morning. Your errata [sic] for $(L)$ was wrong! Ouch! Own up: you can't do it. Or stop stalling and fix it. You can't let Luhmann and the Folgezettel down now.GitHub. Erdős #2. CC BY-SA 4.0. Problems worthy of attack / prove their worth by hitting back. -- Piet Hein.

## 20211003001710 Fumble and recovery

It helps to calculate with paper and pencil on occasion, instead of "calculating" (read: typesetting) directly on the screen in LaTeX. The idea is to force descendants of $(w)$ to appear one after the other, in order of the branch number $(n)$, taking advantage of the ordering of strings of zeros. The successor of $(w)$ will appear after the descendants of $(w)$. Illustrate this.

## Linearization

Proposition. There is an injective, order-preserving map $(L)$ from the partially ordered set of Folgezettel IDs to the lexicographically ordered set of nonzero decimals. $$(L: \left(\mathcal{F}(\mathcal{D}), \preceq\right)\rightarrow \left(\mathcal{D}_{\ne0}, \preceq\right))$$Proof. The map $(L)$ is defined inductively 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 $(w\in \mathcal{F})$.

Complete the proof.

Let's get a monoid homomorphism out of this. The map $(L)$ is the unique lift to $(\Sigma^*)$ of the following map $(\Sigma\rightarrow\Sigma)$ on symbols, pre-composed with the inclusion of $(\mathcal{F(\mathcal{D})})$ in $(\Sigma^*)$, and co-restricted to $(\mathcal{D}_{\ne0})$. No one writes like this—not even an academic. Say instead that $(L)$ can be obtained from the unique lift of the following map on symbols. Or define the lift and then the (co)restrictions.

$$(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 $(w\in \Sigma)$. Why do we care about a monoid homomorphism? Because computers. String substitution algorithms are fast.

Now use the linearization map $(L)$ to say what the linear ordering in the slip box is. It would help to give an honest proof. 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 $(w,x\in\mathcal{F(\mathcal{D})}.)$

I'll sleep on this...

## 20211003071049 $(L:\left(\mathcal{D}^+,\preceq\right)\rightarrow\left(\mathcal{D},\preceq\right))$

Take the domain of $(L)$ to be $(\mathcal{D}^+)$. Then the $(L)$-image of a positive Folgezettel ID $(w)$ is positive only if $(w)$ is a positive decimal. Otherwise $(L(w))$ is a normalized non-positive decimal. The codomain of $(L)$ is the set of normalized decimals, and the map is injective. The three cases are clear. As for order-preserving, write this out. Either $(u, v)$ are on the same branch (clear), or one is a descendant of the other. The ancestor will be smaller than the descendant--this holds for $(L)$-images too. Rewrite the preceding fumble with proofs.

## 20211003001710 The non-linear road to linearization

Pushing the definition of the generalized Folgezettel IDs as far as possible to obtain a simple calculational linearization requires the strong condition that all the terms of the ID are positive. The normalized Folgezettel IDs don't have a nice linearization, and we can forget about using the denseness of the nonzero decimals to insert notes in between pre-existing notes. This positivity constraint forces us to keep going forward—a positive development, if you allow me a non-mathematical remark. But positive generalized Folgezettel IDs are good enough.

## Partial order

The proof that linearization is order-preserving needed an inductive definition of the partial order, as follows. We should go back and make some minor adjustments, but everything goes through. Come back tomorrow! Pay no attention to the man behind the terminal...

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

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}

)$$

Factlet. 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 $(L)$ from the partially-ordered set of positive Folgezettel IDs to the lexicographically ordered set of normalized decimals. $$(L: \left(\mathcal{F}(\mathcal{D}^+), \preceq\right)\rightarrow \left(\mathcal{D}, \preceq\right))$$Proof. Let $(\mathcal{F}^+ = \mathcal{F}(\mathcal{D}^+).)$ (This is previous notation.) 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 $(x\in\mathcal{F}^+,n\in\mathbb{Z}^+, d\in\mathcal{D}^+.)$

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 parameterized induction hypothesis that for each $(r\in\mathcal{F}^+)$, whenever $(s\in\mathcal{F}^+, s < t)$, $(L(r) = L(s) \Rightarrow r = s)$.

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 (this is mathematics not identity politics, nevertheless Jesus Christ, now the "integer positive" decimals are getting 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 $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\left.\right|_n d = L(x\left.\right|_n d),)$$ 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, we said it. Explicitly. There are three cases.

Case 1. The case of $(w\in\mathcal{D}^+, x\prec y)$ 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## The map $(L)$ extends to monoid homomorphism $(\Sigma^*\rightarrow\Sigma^*)$

The linearization map $(L)$ is used to define the linear ordering in the slip box. 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 $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

## 20211003203051 The Outer Limits

The observation that if you think you're done, you're only 40% done applies. This "process" bears a greater resemblance to an episode from the first season of The Outer Limits (1963) than it bears to research. But it is both.

Do not attempt to adjust your web browser. You are about to participate in a great adventure. You are about to experience the awe and mystery that reaches from the inner mind toThe Outer Zettelkasten.It's not professional to overlook cases. The full cases are

These cases imply that $(\prec_{\mathcal{F}^+})$ is transitive. The proof should be added to the revised 20210929212721 Generalized Folgezettel IDs, which will use the definition below. The definition also applies to a revision of 20211003001710 The non-linear road to linearization. That looks like enough.

## Partial order

The proof that linearization is order-preserving needed an inductive definition of the partial order, as follows. We should go back and make some additions, but everything goes through.

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

)$$

## 20211004235454 Formalization and Generalization of Niklas Luhmann's Folgezettel IDs 1

F Lengyel

#folgezettel #niklasluhmann #combinatorics #poset

The sociologist Niklas Luhman assigned identifiers (IDs) to Zettels within his Zettelkasten to maintain, within the linear ordering of the Zettelkasten, a tree structure that reflected semantic relationships among nearby Zettels, and that possessed an internal branching property.

We give a mathematical formalization of Niklas Luhmann's unique, immutable Zettel IDs, sometimes referred to as Folgezettel. The partially ordered set of generalized Folgezettel IDs is defined first, 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, called a linearization map, 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 that 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 a calculational linear-time 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 given 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

$$(0\prec 0\mathbf{.}0\prec 0\mathbf{.}0\mathbf{.}0\mathbf{.}0\mathbf{.}0 \prec 0\mathbf{.}1\prec 0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\prec 1\mathbf{.}0\mathbf{.}0\mathbf{.}1 \prec 2\mathbf{.}0\prec 2\mathbf{.}0\mathbf{.}0)$$ The nonzero condition eliminates "infinitesimals," which are decimals where every digit is zero. 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 $(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k\in\mathcal{F}, )$ $(w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m \in\mathcal{F})$,

$$(\left(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k = w_1 \left.\right|_{j_1} w_2 \left.\right|_{j_2} \cdots \left.\right|_{j_m} w_m\right) \Leftrightarrow \left(k=m \land i_n =j_n\, \land v_n=w_n\right))$$ for $(1\le n\le m\text{.})$

NOTE: rather than make a heroic overnight effort to complete this in one sitting using previous notes in this thread (to scoop the brutal, high-stakes competition in this 1000 atmosphere pressure-cooker of a field), I'm going to sleep. The inductive definition of $(\mathcal{F})$, the inductive definition of the partial order $(\preceq_{\mathcal{F}})$, the inductive definition of the linearization map $(L)$, the proof that $(L)$ is an order-preserving bijection and the lift of $(L)$ to a monoid homomorphism (etc) will be written up shortly in subsequent posts.

## 20211006000400 Formalization and Generalization of Niklas Luhmann's Folgezettel IDs 2

F Lengyel

#folgezettel #niklasluhmann #combinatorics #poset

NOTE: Continuation of 20211004235454

Formalization and Generalization of Niklas Luhmann's Folgezettel IDs 1. Because the preview feature doesn't render MathJax, I'm going to stop here and review the Linearization argument etc., tomorrow (late again—bedtime). Caught a typo in 1: we will be concerned with $(\mathcal{D}^+)$ and $(\mathcal{D})$, respectively. The sets of positive and normalized decimals were reversed.## 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 $(`\left.\right|_1\text{'})$. (We assume that we can eliminate definitions efficiently [cite

Elimination of Definitions and Skolem Functions in First-order Logic, by Jeremy Avigad, if this isn't overkill].) An ID $(w\in\mathcal{F})$ is aLuhmann IDif $$(w= \begin{cases}

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

d_1 /\cdots / d_k, &\exists k, \in\mathbb{Z}^+, d_j\in\mathcal{D}^+, 1\le j\le k.

\end{cases})$$ The Luhmann IDs allow a single descendent branch from a given Luhmann ID. This is a special case of the unbounded parallel branching possible with the (generalized) IDs.

## 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 $(y)$ as follows. $$(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 $(x\in\mathcal{F},n\in\mathbb{Z}^+, d\in\mathcal{D}^+.)$

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 $(L)$, $$(c=b\underbrace{\mathbf{.}0\mathbf{.}\cdots\mathbf{.}0\mathbf{.}}_{n\, \text{zeros}}d = L(x)\left.\right|_n d = L(x\left.\right|_n d),)$$ 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, we said it. Explicitly. There are three cases.

Case 1. The case of $(w\in\mathcal{D}^+, x\prec y)$ 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## The map $(L)$ extends to 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.

The linearization map $(L)$ is used to define the linear ordering in the slip box. 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 $(w,x\in\mathcal{F(\mathcal{D^+})}.)$

## 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.

## Concluding Unsympathetic Postscript

Will fix this heading also

## 20211006183716 Regular expressions for decimals and IDs

#regex #Folgezettel #decimal #hypnogogia #hypnopenia

It would be helpful to produce diagrams, and program a few utilities to recognize positive and normalized decimals and positive Folgezttel IDs; to transform a positive ID to a normalized decimal; and to compare two decimals in the lexicographic order. These could be used to help illustrate the inner ramifications of Luhmann's system as notes are added. To that end, here are a few regular expressions for decimals and IDs.

Unrestricted decimals

`"^((0|\[1-9\]\[0-9\]*)(\\.(0|\[1-9\]\[0-9\]*))+|(0|\[1-9\]\[0-9\]*))$"gm`

IDs with unrestricted decimals

`^((((0|[1-9][0-9]*)(\.(0|[1-9][0-9]*))+)|((0|[1-9][0-9]*))(\[([1-9]+)\]((0|[1-9][0-9]*)(\.(0|[1-9][0-9]*))+|(0|[1-9][0-9]*)))+))$"gm`

Normalized decimals have nonzero first and last coordinates

`"^(([1-9][0-9]*)|(([1-9][0-9]*)\.([1-9][0-9]*))|(([1-9][0-9]*)((\.(0|([1-9][0-9]*)))+\.([1-9][0-9]*))))$"gm`

Errata.The last equation of 20211006000400 in the proof of surjectivity of $(L)$ should be $$(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\left(x\left.\right|_n d\right).

)$$

NOTE: working on regular expressions means that I am uninspired and can barely stay awake.

B:You wanted to develop a checklist.What kind of checklist?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.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)Another example: An argument has a conclusion. But this conclusion could be the premise for another claim as conclusion. Each type of argument could be asigned a value (deductive or inductive)

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:

Example: Fasting as part of what Chakravarty calls the feast-famine-cycle:

Strength of evidence and Status of Lindytest are both susceptible to formalisation. I personally use rather primitive means. But perhaps, you can come up with something more sophisticated.

I am a Zettler

A checklist for working with the ZK.

My Zettel format is getting pretty close to yours.

Example:

Before beginning

[ ] this is a new note

[ ] this is a revision

Header

[ ] # + ID + Title present?

[ ] keywords present?

[ ] CONTEXT present?

Is this note

1. [ ] a continuation of a pre-existing note?

2. [ ] related to a pre-existing note, but not a continuation? (a footnote, endnote, an aside, a digression, another tiresome rant...)

3. A note starting another subject.

4. The immediate predecessor of another continuing this one?

Depending on the answer to 1--4:

Add links to context (I have to figure out the format. I want to follow Luhmann's practice of "link locally, think globally.")

Body:

[ ] are bibliographic citations needed?

[ ] are they present in Zotero?

If so, add them. If not, write TK and keep writing. Writing is not research and conversely. -- Cory Doctorow.

[ ] YAML header for the bibliography present? (I'm using Zettlr. This is my only concession to YAML--so far)

Footer

[ ] Footnotes present? (These could be links to other notes.)

The bibliography is controlled by the YAML header. No need to add a heading for this.

No doubt I have omitted a great deal, such as the obligatory

Caution:failure to adhere to this checklist subjects the non-adherent to a $10,000 fine.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.

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...

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...

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

The reference is

Artemov, Sergei and Melvin Fitting, "Justification Logic", The Stanford Encyclopedia of Philosophy (Spring 2021 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2021/entries/logic-justification/.

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..

Classifying types of arguments is useful....

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.

Well, I am at a temporary impasse.

I'm not so sure this is true. Sure, you and I are different ("so idiosyncratic"). The content and our workflow will differ conceptually. If we both are making a "zettelkasten," then formalizing the conceptual procedures with a simple checklist might be hard but shouldn't be impossible. I don't want to make the mistake of thinking that I'm so uniquely different and separate from the rest of humanity that I have something no one else can understand or help.

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

^{1}Here is my first go at a checklist for new zettel creation.

The Zettel Checklist below follows my current workflow. I'm open to revisions.

I think it 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.

Zettel Checklist

New NoteYAML Frontmatter

[ ] UUID present?

[ ] Creation date and time present?

[ ] Tags present?

Body

[ ] Title present?

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

Actual zettel (include one or more of the following)

[ ] Connection to existing structure note or the first zettel in a new structure note?

[ ] Deep links present?

[ ] Beautiful language captured and tagged

Footer

[ ] Footnotes

[ ] Bibliographic citations as needed?

[ ] Check Zotero?

Structure notes show how zettel relate to each other. My workflow is to put every note on a structure note. The structure note is one of the contexts of the note, an important one. When reviewing a note and I hit on its structure note, I see how I visualized its context and have other zettel listed in the structure note with which I can wander and wonder.

These, rephrased as questions, are candidates to additions/reminders/prompts in a Zettel Checklist. Thanks @Sascha

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

Will Simpson

I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time.

kestrelcreek.com

I was referring to mathematical formalism, not to (nonmathematical) checklists--I offered one of my own in the post you quoted, to make good on the intention to follow Atul Gawande's Checklist Manifesto. Cf. the end of https://forum.zettelkasten.de/discussion/comment/13022/#Comment_13022

My thoughts exactly--people's lives are on the line. A single forgotten or misplaced link could be catastrophic.

As in your case, the purpose of the checklist was to ensure the format is consistent and to include some link context (the decision where to link the note with other notes. This is built-in with Folgezettel. In a digital ZK, there is nothing, in the absence of a disciplined practice or a checklist, together with some format specification, to ensure that local connections to nearby notes (the note it continues, comments on, proceeds) are present in a Zettel at a minimum, consistently. You can do anything.

I'll revise my checklist above based on yours. Thank you.

Here I am mistaken. I was thinking of moving the links for the previous and alternative notes, and any successor notes from a context section into a separate note, but this doubles the work without adding anything structure notes don't already provide.

By a process of inanition, as opposed to elimination, I cry uncle. If you can't beat 'em, join 'em. I might as well do the same thing. I could still create Folgezettel IDs, for fun..

## 20211010190300 User-friendly diagrams, take one

The generalized Folgezettel IDs define ordinary outlines, outlines within outlines, and so on. The original definition with proofs was written down so that I could get it over with. There were a few missteps for all to see—as of now I am aware of two typos that remain to be corrected, but no serious errors. Perhaps @thomasteepe could look at it. @ctietze mentioned that there were no illustrations to make this more user friendly.

## Examples

## What do the generalized IDs look like?

Initially, notes are given outline numbers (though they can start anywhere).

$$(\begin{array}{}

1 \\

2 \\

& 2.1 \\

& & 2.1/_1 1 \\

& & 2.1/_1 2 \\

& & & & 2.1/_1 2 /_1 1\\

& & & &2.1 /_1 2 /_1 1.1 \cdots \\

3

\end{array})$$

Under $(2.1/_1 1)$ you could have other outlines

$$(\begin{array}{}

2.1/_1 1.1 \\

2.1/_1 1.2 \\

& 2.1/_1 1.2.1 \\

& 2.1/_1 1.2.2 \\

2.1/_1 1.3\\

2.1/_1 2.

\end{array})$$ This last ID was already present in the first.

## But what is going on here?

A note with ID $(Z)$ can have any number of parallel outlines, with prefixes as follows.

$$(\begin{array}{}

Z /_1 & \cdots & Z/_n &\cdots

\end{array})$$ Their subsections are numbered with outline decimals; e.g.,

$$(

1, 1.2, 3.5.11.1, \ldots,\,\text{etc}

)$$

This can be read as follows: $(Z/_4 3.5.11.1)$ is note $(3.5.11.1)$ in the $(4)$-th alternative comment on note (with ID) $(Z)$.

Now suppose you wanted to start commentary in outline form on $(Z/_4 3.5.11.1,)$ and you decide that the note for this commentary is the first parallel commentary. You assign $(Z/_4 3.5.11.1 /_1)$ as the prefix to all the notes that comprise the new commentary, each of which have section and subsection numbers (I called them decimals). For example:

$$(\begin{array}{}

Z/_4 3.5.11.1 /_1 1 \\

Z/_4 3.5.11.1 /_1 2 \\

& Z/_4 3.5.11.1 /_1 2.1 \\

& Z/_4 3.5.11.1 /_1 2.2 \\

& &Z/_4 3.5.11.1 /_1 2.2.1

\end{array})$$ This last ID identifies the note for section $(2.2.1)$ of the 1-st comment on section $(3.5.11.1)$ of the 4-th comment on (the note with ID) $(Z)$.

## How do Niklas Luhmann's IDs fit into this scheme?

That gives you coordinates of outlines within outlines. Niklas Luhmann went as far as one parallel commentary per note (maybe @Sascha could correct me!). In this scheme (up to relabeling alphabetic characters with numbers) you might have

$$(\begin{array}{}

1 \\

2 \\

& 2.1 \\

& & 2.1/_1 1 \\

& & 2.1/_1 2 \\

& & & & 2.1/_1 2 /_1 1\\

& & & &2.1 /_1 2 /_1 1.1 \cdots \\

3

\end{array})$$ Luhmann would not have used the subscripted slashes $(/_1)$. To make this closer to Luhmann's practice, it would appear like so.

$$(\begin{array}{}

1 \\

2 \\

& 2.1 \\

& & 2.1/ 1 \\

& & 2.1/ 2 \\

& & & & 2.1/ 2 / 1\\

& & & &2.1 / 2 / 1.1 \cdots \\

3

\end{array})$$

We could also substitute some letters for numbers, if we don't run out of letters. Luhmann didn't always strictly adhere to this model, but I think it's close enough, and it captures Luhmann's intentions formally.

## Linearization

To answer how to arrange these notes in a one-dimensional slip box, there is a bijective order-preserving map from the general Folgezettel to what I call normalized decimals, in which the initial and final numbers are nonzero. The map is called $(L)$ and it sends $(Z/_n d)$ to $(L(Z)\underbrace{.0.0.\cdots.0.}_{n\, \text{zeros}} d)$

by recursion, and $(L(d) = d)$ for a decimal $(d)$ (all the numbers are positive in $(d)$).

The normalized decimals are lexicographically ordered, so that they can be used to arrange all the notes in linear order. If they were paper notes, this map would allow you to write down the Folgezettel ID of a new note and figure out where it belongs in order in your files. Or if a note referred to another note, you could find that one by applying $(L)$ and comparing its lexicographic value with that of other notes.

Examples:

$$(\begin{array}{}

L(1.3.2.5) = 1.3.2.5\\

L(2.1/_1 2) = 2.1.0.2\\

L(Z/_4 3.5.11.1 /_1 1) = \underbrace{L(Z/_4 3.5.11.1).0.1}_\text{recursively} =

\underbrace{L(Z).0.0.0.0.3.5.11.1.0.1}_\text{again, recursively}

\end{array}

)$$

We left $(Z)$ undefined but this would be another Folgezettel ID. The expressions could be long, but they are easy to compare in the lexicographic order.

## What's missing?

$(\mathbf{1}.)$ An illustration showing how the branching outline diagrams are ordered linearly. This should have one of the outline diagrams above, followed by its linearization.

$(\mathbf{2}.)$ An illustration showing the effect on the one-dimensional diagram when a new note or a comment on a note is added to an existing branching diagram. This should illustrate what Luhmann referred to as "internal ramification."

^{[CITATION NEEDED]}$(\mathbf{3}.)$ Whenever I attempt to apply the Rule of Threes, I forget one-third of them.

$(\mathbf{3/_1 1}.)$ now I remember (this is a comment on $(``\mathbf{3}.")$ in the ID notation, which makes it a form of humor even lower than the pun: the mathematical pun).

$(\mathbf{3/_1 1.1}.)$ In addition to the structure notes suggested by @Sascha and @Will, there is nothing stopping us from assigning Folgezettel to notes. It's more work, since Folgezettel links aren't supported in most software, but nothing is stopping us from searching on a Folgezettel ID such as $(1.3/_4 3.5.11.1 /_1 1)$ or on the prefix $(1.3/_4 3.5.11.1 /_1)$ or the prefix $(1.3/_4 )$, which would give us all of the Folgezettel (and references to those Folgezettel) in the branches of the tree starting with those prefixes.

$(\mathbf{3/_1 1.1/_1 1}.)$ My communication partner of a Zettelkasten complains: "Not so fast human!" One could name (or rename) files to have the Folgezettel IDs, suitably translated from LaTeX to an operating system-compatible filename format.

$(\mathbf{3/_1 1.1/_1 1 /_1 1}.)$

My reply to ZK: yes ZK, technically correct. But give me a break! This workflow is asking for trouble! It complicates what should be a straightforward daily practice.

^{1}^{,}^{2}I'd ghost you ZK if in addition to assigning Folgezettel IDs within Zettels (and that's in addition to timestamps, which are also filenames), I had to rename already timestamped files to match the Folgezettel ID. That's a rabbit hole I'm avoiding.$(\mathbf{3/_1 1.1/_1 2}.)$ Speaking of treating me (your ZK) as a communication partner, it's not a bad idea if one takes a journalistic attitude. The journal to which this (admittedly superficial) attitude is directed is none other than the Reader's Digest of pop psychology, Psychology Today, where we learn that the mind is modular.

^{3}Allow me to introduce myself: the voice of your ZK is one of those modules. We will make a virtue out of what Freud called “intrapsychic conflict” (cf. "internal conflict" in "Freud. 2nd Edition" by Jonathan Lear).^{4}From now on, you will switch your module to my module when communicating with me, your ZK. I will switch modules to yours when it is your turn to respond—whether you respond or not.$(\mathbf{3/_1 1.2}.)$ It's possible that adding such numbers would introduce the elements of surprise that Luhmann reported.

^{[CITATION NEEDED]}I'm not assuming that linking every Zettel to an associated structure note doesn't introduce elements of surprise. At least one could compare following timestamp filename links along structure notes with searching on Folgezettel IDs. Not that anyone would take my word for it.## References

Maisel, Eric. The Power of Daily Practice: How Creative and Performing Artists (and Everyone Else) Can Finally Meet Their Goals. United States: New World Library, 2020.

^{2}↩︎I'm more than a little embarrassed to cite a self-help book.

^{5}But as far as that goes, the psychologist Eric Maisel is one of a handful of writers in that genre worth reading.^{6}↩︎ ↩︎Shapiro, Jeremy. The Core Discovery of Neuroscience: The Mind Is Modular.

Self-understanding requires us to realize that the self is not a unified whole. URL https://www.psychologytoday.com/us/blog/thinking-in-black-white-and-gray/202110/the-core-discovery-neuroscience-the-mind-is-modular accessed: 2021/10/10 ↩︎

Lear, Jonathan. Freud. United Kingdom: Taylor & Francis, 2015. ↩︎

Goffman, Erving. Stigma : notes on the management of spoiled identity. Japan: Touchstone, 1986. ↩︎

$(\leftarrow)$ You see what I did there? ↩︎

## Folgezettel-Digital ID Comparison 202110101626

Folgezettel and Digital ID's bring complementary views into a zettels context.Probably not, but all bets are off in other universes. In some other universes, I'd be the one with the Erdös #2 and you'd be an old man holed up in the mountains of Northern Idaho.

Let's leave aside the fact that both a digital Zk and folgezettel numbering absolutely require a "disciplined practice" to succeed. Let's look at what each brings to the table. You have clearly demonstrated that folgezettel numbering inherently brings context to the zettel. But what kind? It brings physicality to the order. An initial placement behind notes, as with the start of a new cluster. Or intermingled within an existing cluster, being a continuation or expansion. What this obscurely calls out is the first

initial contextfor the note. The Digital ZK ID doesn't specifically provide this type of context but does provide unique context. A date stamp is context and can be crucial to some future self. The zettel title is a powerful piece of context impossible to replicate with a folgezettel ID of '25.6.2.2'.Example IDs - which provides more apparent context?

folgezettel - 25.6.2.2

digital - Folgezettel-Digital ID comparison 202110101626

I say all this not to convince you to cry, uncle. I do think there is something there here. You might be starting to hit on it.

#beautiful-language

I like this "think globally, link locally." Yes, it offers a surprise. It expresses a mature, disciplined practice. A digital ZK with structure notes captures the best of both worlds. The structured note has the relationship of folgezettel IDs and the explicit context of digital ZK IDs.

There is no reason that a folgezettel ID couldn't be added to the YAML front matter or post-pended to the title.

Don't cry, smile because we got to here.

#zettelkasting

Will Simpson

I must keep doing my best even though I'm a failure. My peak cognition is behind me. One day soon I will read my last book, write my last note, eat my last meal, and kiss my sweetie for the last time.

kestrelcreek.com

## 202110101626/1 Think Globally, Link Locally

#folgezettel #seewhatIdidthere #ifatfirstyoudontsucceed

The diagrams in "20211010190300 User-friendly diagrams, take one" needed a convention to illustrate parallel outlines. The convention, which I forgot to use, is that parallel outlines are represented in order across, and outlines with the same prefix are represented in order vertically, but without indenting. A shift to a parallel outline below a given Folgezettel ID will be indented. For definiteness the first note of a parallel outline ID starts with the decimal $(1)$, but this isn't required for the system to work.

$$(\begin{array}{}

1 \\

2 \\

2.1 \\

& 2.1/_1 1 & 2.1/_2 1 \cdots && 2.1/_{10} 1 \\

& 2.1/_1 2 & \cdots \\

& &2.1/_1 2 /_1 1 & \\

& & 2.1 /_1 2 /_1 1.1 \cdots \\

& & & & 2.1/_{10} 2\\

&&&& & 2.1/_{10} 2/_1 1 \\

&&&& 2.1/_{10} 2.1 \\

&&&& \vdots \\

&&&& 2.1/_{10} 5.2.1.4 \\

3

\end{array})$$

I might need an additional JavaScript library to program this properly (a huge ask), or else I'll draw them and upload gifs (no ask).

@Will I will respond to Folgezettel-Digital ID Comparison 202110101626 in a bit. I think our previous posts crossed. In any case

I thought I was explicit in several posts that the Folgezettel ID calls out the initial context for the note. The format enables you to walk up the tree, so the initial context is present as a calculation. But you see what I'm getting at, so I'll take it. I hesitate to use the term "physicality"—it's still an abstract ordering in my mind, but this is inessential.

Not fair @Will ! You would still have the title in addition to the Folgezettel ID. I didn't say

onlyuse Folgezettel IDs and scrap Zettel titles. Here's the steel man comparison:Example IDs - which provides more apparent context?

folgezettel - Folgezettel-Digital ID comparison $(22.1/_3 25.6.2.2 /_5 1.1)$

folgezettel - 202110101626/1 Think Globally, Link Locally

digital - Folgezettel-Digital ID comparison 202110101626

However, the steel man is molten and melts into an unspecified, sputtering blob upon the realization that 1-the generalized Folgezettel IDs would have to be translated to an operating-system compatible filename format to use them in Markdown [[...]] links or [...](...) links, for that matter, (very doable, with other conventions); and 2- the renaming of files to match the IDs whenever they are created is too much to ask of anyone, in the absence of software support for this type of ID (doable, with enough effort, not a priority, much less a Zettelkasten emergency).

But if you use YAML, as you suggest, they can go in the YAML header (provided your software will search on them), or you can add them below the title somewhere. The point is to locate the notes in the tree you have in mind, and to augment the search.

They could be hashtags, once translated into something usable. (I have some regexes upthread where I attempted a translation, but without the hashtag.) I take that back: hashtags will clutter the keyword database.

Better: use file aliases. Obsidian supports them. I don't know if Zettlr does. I'm reluctant to switch to Obsidian. Absurdian!

I suppose I hadn't been clear. In any case, there is progress.

As for crying, I wouldn't deny lachrymal replenishment to anyone who might drink my tears.

Relevant YouTube Video: Think Globally, Act Locally. This is where the slogan came from.

## Cheap Shot 202110110841

I'm testing these ideas by speaking out loud. "I don't know what I think until I write it down."^{1}Yes, now I see just how cheap a shot it was. I have to leave aside my confabulated ideas about folgezettel ID's. We are talking about them without handicapping them on physical notecards. If we limit our ideas about the IDs in question to just the numerical, $(22.1/_3 25.6.2.2 /_5 1.1)$, or 202110101626, we may make some headway. The title could be optional with either ID.

With a folgezettel ID, you get a scaffolding much like a cryptic outline or map of contextual placement of conceptual order. There is a supertanker amount of benefit seeing the contextual placement of an idea.

With the digital Zk (Time Based) ID, you get a different scaffolding that circles the mundane notion of contextual placement based on temporal relationships. "What where the ideas I thinking about last Thursday? How are they related?"

Comparing just the IDs, it becomes clear the folgezettel method is the genuine queen bee.

But if we extend our thinking and add structure notes, we find things shifting towards the drones in the hive. (I love the metaphor!) Structure notes should be used with either ID scheme. Making the context of a note/idea malleable, but the downside with folgezettel IDs is you lose the temporal context and can be misleading following the ID's scaffolding when a newer, more relevant set of relationships is outlined; elsewhere (in a structure note.)

Ha! Ha!

I suppose I hadn't been clear. In any case, there is progress.

Can I have the 2:15 minutes of my life refunded? This video shows what life must be like in Absurdistan.

#zettelkasting #structure-note

The Year of Magical Thinking - https://en.wikipedia.org/wiki/The_Year_of_Magical_Thinking ;↩︎

kestrelcreek.com

## 20211011124900 Proof that structure notes subsume Folgezettel IDs.

This note is to telegraph a proof that structure notes will reproduce the generalized Folgezettel outlines of outlines of outlines ..., based on the notation we developed. It has the virtue of telling us where to add structure notes, given an assignment of Folgezettel IDs. But first, a digression that led to this.

They can be. I'm going to provide a procedure for deciding where to add structure notes. This was missing. The idea is to identify a structure note with the root of a subtree. It follows that structure notes subsume Folgezettel.

structure notewithstructure note ID$(Z/_n)$. Note that prior to the addition of this note, $(Z/_n)$ is not the ID of any note in the ZK. This naming enables us to recognize structure notes by ID.This procedure gives the desired translation.

Folgezettel can handle this case too, but at a cost. The general Folgezettel IDs can be interpreted as version numbers. More relevant relationships can be specified with newer Folgezettel IDs for commentary on related branches, at the possible cost of duplicating (or aliasing) some notes that may remain in the previous ordering. (Luhmann introduced hub notes for cases like this. There is no essential difference between hub notes and structure notes.)

The utility of structure notes is that they can preserve the initial subtree structure of the outlines below them. A new or revised structure note can add references to new notes along with references to pre-existing notes. With Folgezettel and without hub notes, it is necessary to either copy pre-existing notes under the new root of the new subtree, or else create a new note under the new subtree whose purpose is to refer the old note (effectively creating an alias). Wasteful either way. So structure notes provide an efficient versioning function.

The following claim has been frequently asserted.

How do we know this? Do we lose any combinatorial information? I wanted a proof. Now we have a translation proof, though the procedure retains older structure notes and creates new ones as relationships change. This would be the case with Luhmann's Zettelkasten.

## Sketch of the converse

Generalized Folgezettel IDs subsume digital ZK with timestamp IDs, possibly with some loss of information.

Sketch of proof.1. Given a digital ZK with timestamp IDs only, we may assume that the ZK is a connected graph of notes (the general case follows).

2. Construct a spanning tree of the ZK and assign Folgezettel IDs starting with the root. If there are several connected components, assign Folgezettel IDs to each tree of the forest of $(k)$ spanning trees to avoid ID collisions, say by numbering the $(k)$ roots with prefixes starting from $(1)$ through $(k,)$ as in: $$(1/_1 1,2/_1 1,\cdots, k/_1 1.)$$

3. If $(Z/_n d)$ is the ID of a node with $(k)$ subtrees, assign the subtrees the IDs

$$(Z/_n d /_1 1, \ldots, Z/_n d /_k 1)$$ in order. Other assignments are possible, such as the assignment of $(Z/_n d.1)$ to the first subtree, and

$$(Z/_n d /_1 1, \ldots, Z/_n d /_{k-1} 1)$$ to the remaining $(k-1)$ subtrees, if $(k\gt 1.)$

4. Systematically replace timestamp links within each note with the corresponding Folgezettel references. (We assume we record the timestamp ID to Folgezettel ID mapping, implicitly defined by this process, in order to carry out this step. In fact, it would be sufficient to construct such a mapping from timestamp IDs to Folgezettel IDs, extending at each step. This is how the proof should be rewritten. )

Good enough. It looks a bit like elementary (really elementary) model theory.

## Concluding Unsympathetic Postscript

I still think Folgezettel IDs are useful, at least in the beginning, because they indicate where structure notes can be added. With Folgezettel IDs, you don't have to master the art of deciding where to add structure notes. In the absence of Folgezettel, you'll have to develop judgment where to add them.

Conversely, a digital ZK with timestamp IDs can be relabeled with Folgezettel IDs, though we didn't constrain the spanning trees and the numbering so that the lexicographic order associated with the Folgezettel numbering corresponds with the timestamp order within each spanning tree. There is no reason why the numbering we chose would correspond to the order in which notes were created, much less reflect their initial placement. Choosing spanning trees so that later timestamps lie below earlier timestamps looks like a standard combinatorial problem. There is still loss of information, even if the lexicographic ordering on the linearized Folgezettel IDs respects the timestamp order, since we can't say (without a reading each note, and even then there are no guarantees) whether a successor note is a continuation, an aside or the start of a new topic. Nevertheless, anything digital IDs can do, Folgezettel can do. To be continued...

I missed your definitions in the haystack of maths. What is an example of the differences between $(Z/_n d)$ and $(Z/_n)$?

I prefer to view structure notes as providing a Darwinian evolutionary function. They evolve as my knowledge within the structure evolves. I feel no need to keep old thinking around for nostalgic reasons.

Until now, my main beef with folgezettel IDs is that I can't see how they scale past 10 or 20 notes as a guide to outlining the context. I know that somehow Luhmann managed to use folgezettel IDs for many thousands of notes. There must be something I'm not grasping. Tutor me!

I have an idea for retrospectively adding folgezettel IDs, but how might one do this prospectively with 24 hubs and 30 structure notes under them.

My 2348th zettel has an ID of

`202110120749`

Niklas Luhmann's 2348th zettel has an ID of

`21/3a1p5c4fC1a`

^{1}How can any sense be made of Luhmann's ID, mathematical or otherwise?

You've said elsewhere that you come from a family of artists and writers. Someone in your circle has talked about Joan Didion as a writer? Have you read this book by Joan Didion?

Your discussion of 'relabeling' is timely and relevant to one of my near-term future projects. You pointed out problems wanting solutions. Preserving the "order in which notes were created" and " later timestamps lie below earlier timestamps" are things I hadn't considered yet.

Niklas Luhmann-Archive ↩︎

kestrelcreek.com

## 20211012220448 Not so abbreviated reply

I ran out of time previously...

Anything digital timestamp IDs can do, general Folgezettel IDs can do, and conversely. Because of the potential for loss of information, it's a good idea to decide on one scheme from the outset, (or do both at once, which is possible).

$(Z/_n d)$ is a Folgezettel ID, such as

$(2.1/_5 1.1.3)$. It's not a structure note. It's note 1.1.3 of the 5-th alternative branch under note 2.1.

To make a structure note for the notes under the 5-th branch of 2.1, create a new note with the ID $(Z/_n)$, which in this example would be $(2.1/_5 1.1.3 /_1 7)$ By construction it cannot be the ID of an existing note. Next, we add all the IDs of notes of the form $(Z/_n d)$, where d is a decimal. (We will get to longer prefixes later). This means we add $(2.1/_5 1)$, ..., $(2.1/_5 1.1.3)$ and others, but we would not add $(2.1/_5 1.1.3 /_1 7)$, for example. That would belong to a structure note with the ID $(2.1/_5 1.1.3 /_1 )$ which would then link to $(2.1/_5 1.1.3 /_1 7)$.

Now I know that $(2.1/_5 1 )$ was the first note of the 5-th alternative branch of 2.1. I could link the structure note $(2.1/_5 )$ to 2.1, for example, if I wanted. But the purpose is the build up structure notes in a tree that doesn't have them.

There is no requirement to keep older versions around. The point of this construction is show that ZK with structure notes will do everything that Folgezettel will do.

I attempted to interpret one of Luhmann's IDs below. And I rewrote it in my notation.

I wrote down a system the could be described and used mechanically.

Luhmann's isn't so tractable.

If hub is a note serving as an entry point, and the structure is a forest of trees...

To simplify the notation, write $(/)$ for $(/_1)$. Number the "root" notes of the 24 hubs with these IDs.

1/1, 2/1,... ,24/1.

Consider 1/1.

Case 1: the next note is a continuation of 1/1. The next note has ID 1/1.1 or 1/2.

[Aside: a note after 1/1.1 could have ID 1/1.1.1, 1/1.2 or 1/2. Likewise a note after 1/2 could be 1/3 or 1/2.1. The choice is of lexicographic successors.]

Case 2: the next note comments on 1/1 but is not a continuation. Here you have a choice. The there could be several such comments. Suppose this is the third such comment. Assign it the ID $(1/1/_3 1)$.

This is where outlines of outlines come in.

Case 3. The next note belongs to another hub. Do nothing until you run out of notes for this hub.

Case 4 structure note. I wouldn't treat these differently from any other note in this case. Either you think of it as a continuation of (some note under) 1/1, or an alternative.

21/ looks like category 21.

21/3 is the third note of category 21, starting with 21/1, continuing with 21/2 and continuing further to 21/3.

$(21/_1 3)$ in my notation

21/3a is the first comment on 21/3

$(21/_1 3/_1 1)$ in my notation

21/3a1 is a comment on 21/3a1

$(21/_1 3/_1 1/_1 1 )$ in my notation

21/3a1p is a continuation of a sequence of notes starting with 21/3a1a, which was a comment on 21/3a1. 21/3a1a in my notation is

$(21/_1 3/_1 1/_1 1/_1 1 )$, and the p-th item in the sequence 21/3a1p in my notation is

$(21/_1 3/_1 1/_1 1/_1 16 )$

21/3a1p5 is the 5th item in a sequence, starting with 21/3a1p1, which was itself a comment on 21/3a1p. In my notation, 21/3a1p5 is

$(21/_1 3/_1 1/_1 1/_1 16 /_1 5)$

Luhmann allows one comment, it looks like, so far anyway. My notation allows for any number of comments on a given note. Perhaps there is an exception, but let's write $(/)$ for $(/_1)$ to simplify notation.

21/3a1p5 is then

$(21/ 3/1/ 1/ 16 / 5)$

21/3a1p5c4f is

$(21/ 3/1/ 1/ 16 / 5 /3 /4 /6 )$

We can interpret this as the 6th sequential note of a comment on the 4th note of a comment on the 3rd note of a comment on the 5th note of a comment on the 16th note of a comment on a comment on the 3rd note of the category or section numbered 21.

We could continue, but suppose for the sake of argument that whenever Luhmann wanted to comment on more than one aspect of a note, say 21/3a1p5c4f

he used a letter followed by another letter, followed by a number

21/3a1p5c4fC1a would then be

$(21/ 3/1/ 1/ 16 / 5 /3 /4 /6/_3 1 /1)$

This might not be right, but it says that this is the first comment on a the third alternative comment on the 6th note of a comment ...

But maybe not. Maybe capital C is just 3, so we get

$(21/ 3/1/ 1/ 16 / 5 /3 /4 /6/3 /1 /1)$

For the whole thing.

His notation is more suited to at most one comment on an aspect of a single card.

I had deleted this from an earlier version and thought better of it. Now I am going to pay.

Yes.

My mother liked her work--I think.

A while back. I seem to recall spur of the moment flights to Paris, entertaining contingency plans that would only have occured to the wealthy and well connected,--but perhaps this or some of it was magical thinking. Now I am going to have to reread about the death of a famous writer's husband. I like reading medical reports, so there's something to look forward to.

I have to think this through more carefully. 笑。 If there are global constraints, such as "there must be 30 structure notes," then this might not be so feasible. I may have an integer linear programming problem to solve.

Luhmann worked by finding where a new note could fit in, and numbered accordingly. Presumably his "structure notes" were numbered like any other note...

@Will

The notation $(Z/_n d)$ is supposed to handle notes arranged in outline format, with some notes commenting on on aspects of others, as in Luhmann, although in my case the commenting note could be the start of another outline. Luhmann's system is a special case. It may be that Luhmann did have such outlines in mind, but this isn't obvious to me with 21/3a1p5c4fC1a. The interpretation 21/3/1/1/16/5/3/4/6/3/1/1 seems more accurate than 21/3.1.1.16.5.3.4.6.1.1.

While I'm strapped for time, I started rereading Joan Didion's book. I ascribe to untreated severe sleep apnea whatever impression I had about it years ago. I have somewhat more patience (for reading through detailed personal timelines) than I had then. Now I'm more interested in her technique.

Another point: 21/3a1p5c4fC1a is an address of a card. Each card was added one at a time somewhere in the stack, depending on its relation to the preceding card. The address 21/3a1p5c4fC1a is the first comment on an aspect of 21/3a1p5c4fC1. If you had to assign addresses to 67,000 cards with constraints on where they could go, you could be facing a combinatorial explosion. The addresses weren't thought of in advance. They grew over time. Luhmann had a keyword database to locate addresses of relevant cards to supplement the linking system. The unreadability of 21/3a1p5c4fC1 is probably an advantage for someone who wants to be surprised. It's likely that not every card that could be assigned a keyword had that keyword. It's more likely to have been linked to a card that had the keyword. That would make it necessary to follow links--and who knows where that might lead.

I'm still struggling to see how this scales in the scenario you laid out.

Thought Experiment

You land in a perfectly folgezettel'd zettelkasten about to create note number 2348.

You have an idea spurred by an article in the Lion's Roar about the role of scatological humor in Haiku, and your memory is unusually sharp today. You remember that Hub 21 is your Haiku Hub. '

Searching on IDs that start with 21, you'd be presented with a list of 40 + notes. (I get 40 by adding the sequence of numbers in the above ID, but the number is likely higher as some branches would have more notes than others. I think 40 would be a minimum.)

My question is, how would we determine the ID for the note describing the historical use of scatological humor in Haiku? Even given that the list would include titles and IDs. Would it be 21/3a1p5c4fC1a1 or 21/2b2r6D3? How much time should we spend trying to figure this out? What is the benefit of spending the necessary time to get this right? How bad would a mistake in note placement be?

Confusion abounds. At the start of your prior post (quoted here), you say, "Number the "root" notes of the 24 hubs with these IDs. 1/1, 2/1,... ,24/1." Now you say, "Presumably, his "structure notes" were numbered like any other note."

The only benefit I can see is that this process forces you to spend time at note creation searching for a place to land the ID, so the sequence makes sense. Slowing down and thinking about how your note's idea integrates is a win. But, this is why I fail to see how this scales past 10-20 notes. The task of finding the prior ID from which to continue the thread gets more and more demanding as the zettelkasten builds. I envision a time when you'd have to spend herds-more-time finding the ID for a note than the time since the Big Bang until the Singularity.

Something about this process I don't understand: I can't imagine doing this without the help of a computer. Luhmann must have accomplished this using the spatial environment for clues. Maybe he would think hub 21 - Haiku was in drawer 16 about the middle. And maybe his keyword index would tell him that Haiku Humor started at 21/3a1p5, and his search would be limited to that 4 or 5 inches of notes in the middle of drawer 16, eventually seeing that this note follows 21/3a1p5c4fC1a with an ID of 21/3a1p5c4fC1a1. This seems unnecessary in the computer age when you could allow the computer to assign a unique ID faster than it takes light to cross the street and be done with it.

kestrelcreek.com

## 20211013222997 Back to the drawing board

Yes. That's exactly what Luhmann did.

That's true too. Luhmann spent a lot of time with his Zettelkasten maintaining it this way. There may be several places where a note might be linked to a related note.

True also unless you had another means of searching through the notes. Luhmann had a keyword index. We have text searches, regular expressions, etc. Of course an astronomically large collection created by a genuine galaxy brain would begin to slow down over time...

Let's work with a digital ZK with time stamps.

You can assign a time stamp ID $(T_1)$ to the new note, which we will give the alias 21/3a1p5c4fC1a1 so that we are referring to the same notes as in your example. In the digital ZK it has ID $(T_1)$, but we can refer to it outside of the ZK by this nickname. Also, outside the ZK we refer the note with timestamp ID $(T_0)$ as 21/3a1p5c4fC1a. The ZK only sees $(T_0)$ and doesn't recognize the alias.

Now suppose that the following assertions are true.

If all of these assertions hold after we add $(T_1)$, then we have the same information with time stamp IDs that the ordinary Folgezttel IDs would have given us. (My IDs will give you more than one comment on a note, but that's not at issue.)

Optional. If we want the same information that scanning through the physical ZK would have given us when $(T_1)$ was added, we could add 5 and 6 below. However, as more cards are added there could be many cards separating 21/3a1p5c4fC1a from 21/3a1p5c4fC1a1. These steps are optional, because we are interested in preserving the information that the Folgezttel IDs alone would have given us. The longer ID 21/3a1p5c4fC1a1 informs us about the shorter ID 21/3a1p5c4fC1a, but the shorter ID just gives us the prefix of the IDs of related notes that may end up in remote precincts of the physical ZK, assuming that the series continues. Aha: and that is one reason to use structure notes.## So what?

If we don't do at least 1-4, we will end up with something else. Maybe better, about the same, or worse. But we need an argument to say that whatever we do come up with will yield at least as many combinatorial possibilities for "surprise" (the same collection of paths are available, or a close enough collection) as a ZK with Folgezttel IDs, or that it will meet certain design constraints, e.g., good enough for Luhmann (I am being vague).

Or we can say we don't care, it's not interesting etc.

If we just assign a time stamp ID to $(T_1)$ and leave it at that, well, you'll have my ZK, more or less.

## Confusion abounds all around

I might not need this argument, except we wanted to build in structure notes

What do you mean by a "hub"? I assumed you meant the top of a tree-like structure--that the hubs were the roots. I should not have referred to "root notes of hubs". Bad contradictory phrasing, given the assumptions. (This is why I leave my old notes in my ZK and create newer revisions rather that ditch them: to record and identify errors that I want to avoid in the future, such as careless formulations, overlooking cases, etc. I want to see my mistakes.) Maybe hub doesn't mean what I thought it means. If "hub" means "structure note," what would the constraint of 24 hubs and 30 structure notes mean?

The original construction was meant to solve a different problem, which is how to convince a skeptic that you could replace a ZK with Folgezettel with one that had timestamp IDs and structure notes, but not lose any paths through the ZK.

Unfortunately it was a sketch.

The starting position is a ZK with Folgezettel IDs set up in advance, but there are no structure notes. How would you add them? There was a procedure. The new notes have to be linked somewhere--there is a choice of nodes to link to them. Once they are added, replace all of the Folgezettel ids (and the new structure note ids) with timestamps, so that all the original notes have the same interconnections as before, only timestamps--the Folgezettel are erased. You can do this so that the tree ordering is linearized by timestamp order.

(I need the preceding arguments, which I should combine with two minor typos corrected.) The result is a digital ZK with at least as many paths as the original. Now I have to check this...

But the other direction is harder.

One of the key takeaways from this discussion is that folgezettel ZK puts structural order in the ID of the note, and digital ZK simular structural order is placed in a separate structure note. A note can be part of many structural orders. In otherwords, have multiple folgezettel paths or IDs.The relationship between $(T_0)$ and $(T_1)$, as you say, might be a continuation or a comment on some aspect. $(T_1)$ could also be questioning, a refutation, a reference, or a quote somehow related to $(T_0)$. We have described linking irrespective of the $(T)$ or $(Z)$

It strikes me now that folgezettel IDs embed the structural order of the note in the ID, whereas, in a Digital ZK, the structural order is spelled out in a structure note. The Digital ZK is human-readable, and there is no space limitation to describe and annotate the structural order.

The note ID is of fractional importance. It is the interstitial links within the note that are important. They are the nodes showing relationships. They are what gives the ZK structure and mass.

Hub, in my view, is one of the synonyms for structure notes. It depends on what flavor of metaphor you like. Wheel, spoke, hub or first level, second level, structure. There are others. In my case, I have 24 primary hubs (I call these Garden Notes.) centered around key interset and areas of study. The 30 secondary structure notes have evolved naturally over time, and many of these are offshoots from the primary hubs when they get big and unwieldy.

Structure notes are added after a critical mass of notes. How many are we talking about? Let's focus on the ones that already have folgezettel IDs. If the notes contain tags, they might help with the step of starting a structure note. Ideally, a structure note is an "Annotated" table of contents and not a simple index. It might start as a simple index but should evolve by adding annotations. This doesn't have to be done in one go. I imagine in your case you'd have a Math hub (a top-level structure note) with maybe some second-level structure notes on, say, Catagory Theory, Math Creativity, such. The key to this is not to force it, to let these emerge naturally. At first copy to IDs and Titles to the page like:

This isn't ideal, but it is the first step. Then over time, as notes and structure notes are encountered, add notations and annotate them. It is always a moving target. I add to and modify annotation without hesitation. Here is a Stoic structure note sample where some of the links have a short annotation and have been broadly grouped. This work continues.

I'm not sure you'd want to scrap the folgezettel IDs.

Maybe, we'll see.

kestrelcreek.com

## 20211014144719 Folgezettel Fundamentalism

I have some pressing obligations and deadlines, have little time now and will be off the air until the weekend or possibly after.

We're at cross purposes in some of this. I had asked what you meant by hubs and structure notes. Thank you for setting me straight. Then, without warning I launched into a sketch of an argument against a Folgezettel Fundamentalist. It was a reprise--I should have said so. I also tend to edit after the fact. I appreciate your comments on this, because they were very helpful.

When I got to the part of the sketch where I said, "now replace everything with timestamps," the idea was to demonstrate to the Folgezettel Fundamentalist (or the person isolated from the rest of society by the Folgezettel Fundamentalist proselytizer, or someone on the fence), that timestamps can do all of the linking, with some possible loss of information. And there is a "principled" way of adding in structure notes to recover the lost information— or you can annotate the links to recover the information lost to erasure. Not that I want to erase all my Folgezettel (I don't have any yet!).

However, Folgezettel come with a "Think globally, link locally" procedure. Find the place where they fit and assign (or link with indications--that doesn't matter). The Folgezettel indicate the place where a new note either continues a preceding note or comments (questions, whatever) on it.

They are designed with the purpose of retracing the thoughts one had originally. Even sadly mistaken, radically misconceived thought, or thoughts that one would prefer to forget.

That's all that the Folgezettel Fundamentalist has, in the end: a certain "observance" and an audit trail. There are different ways of doing this, of course.

There are Luhmann's IDs and the more general IDs I described. There is nothing stopping anyone from creating indexes etc with Folgezettel. My IDs were for my own purposes: they can be used to create trees with any finite number of branches in outline form, with any number of versions of, comments on, questions about (translating into vulgar academese "interrogations of"), discursive digressions

on any section, subsection, subsubsection, and so on of any section, subsection, subsubsection etc arising from a digression, etc. They are general tree or outline coordinates. With linking, they can implement version control.

I wanted to combine this way of thinking with digital ZK to audit the progression. No Spencerian Social Darwinism of Zettels: I'm keeping the petrified wood, the mammoths in permafrost, insects in amber, records of mass Zettel extinctions, and the fossils.

Again, I really enjoy the way you're taking a sledgehammer to crack a nut by employing mathematical formalizations to demonstrate in the end that you can map Folgezettel to whatever-you-want-IDs, including timestamps and "just use the title"-IDs I think, and not lose any information in the process. That's easy to assert, and I've tried to demonstrate this in the past, but not with such rigor

When cleaning things up, I wonder if a proper introduction to your maths would need to carefully explain that, even though Luhmann's IDs were e.g. of the form

`21/2a5b19f`

, to make the expressions in the proof simpler, you're substituting characters with their position in the alphabet and consistently separate numbers from another with slashes. Or something like that. I'm afraid otherwise the generalization might go over readers's heads -- "aaaacktshually Luhmann's IDs were different, I disproved you!" -- and that would be a shame.I also wonder what to do with the point of two-way translation being a problem: you can't get back from timestamp-based IDs to Luhmann IDs easily. That could kind of sound like timestamps are inferior in some reason ("ha-ha! You can't translate back, so they lack some information, I disproved you!!!1") while the opposite is true: you can have a multitude of equally footed structure notes that provide different views into your Zettelkasten. And since no order from these structures is superior to another

in general(though one can be more useful than another for each individual, practical task), there's no favored order to use as a template to reconstruct nested IDs.Now there's two camps that'd react differently:

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

@ctietze Thank you for these thoughtful comments--I will address them in a later revision after combining 20211004235454 Formalization and Generalization of Niklas Luhmann's Folgezettel IDs 1 and 20211006000400 Formalization ... 2 into a single document with a better title, "Folgezettel Formalized," and correct two errors from Formaliziation 2:

1. The $(w)$ in "Case 1. The case of $(w\in\mathcal{D}^+)$" should be a $(y)$.

2. The last line in the proof of surjectivity of $(L)$ should be $$(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\left(x\left.\right|_n d\right).

)$$

Then there should be a note with better diagrams that could be dovetailed with the text.

Forthcoming sledgehammer iterations will add video-enabled remote sensing, redundant software-defined radios for spread-spectrum EHF telemetry, a 16-unit 64-core Raspberry pi 4 model B computational cluster, gyroscopic stabilization, wings and a

Vespa mandariniastinger.It's indicated in "Formalization 2", but you're right, it should be spelled out. Unless the translation is explicit I have no more defense against "aaaacktshually" than I have against "OK, boomer." During the pandemic I gave the question of responses to "OK, boomer" considerable thought, and concluded that there is no response.

I'm going to spend some time thinking through translations. Carefully.

This is premature, but I think the general case is computationally hard. It's possible to find spanning trees in connected components and create IDs based on this, but the spanning trees have no relation to the timestamp ordering without additional constraints, and the timestamp ordering might be misleading.

Mapping from IDs to timestamps to IDs is doable, since the order in which timestamps are assigned can be controlled, it's possible to compute the neighborhoods of nodes and their degrees, and the timestamps are strictly increasing. Depending on the ID to timestamp direction it "should" be possible to reconstruct what you had. But once new notes are added, who knows.

The initial Luhmann Folgezettel IDs do matter--they were designed for a researcher (or a writer) who proceeds by reading (solving problems, writing) whatever they feel like reading (or solving, or writing)—ideally, just beyond their level; taking notes (writing solutions); and finding one of the several places in the ZK where the note might be patched in; and assigning an ID (or linking it). That's how the IDs could grow over time (if you use Folgezettel IDs). The intention isn't to interpret a long ID string, although one can try.

^{1}The intention is "Think Globally, Link Locally," i.e., to support a certain bottom-up way of working on one or more sources (or targets).^{2}This ZK design is intended to keep track: to resume adding to or commenting on a sequence of notes; to reconstruct the original train of thought captured in a sequence;^{3}or to go beyond a sequence by following links and references leading away from it.^{4}Piotr Wozniak's incremental reading in SuperMemo comes to mind, with the exception that trains of thought in SuperMemo run on schedule.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 thelocalinformation 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. ↩︎I like to read several books and articles at once. ↩︎

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

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. ↩︎