Links - Zettelkasten

by Alex Nelson, 1 December 2021

When I refer to other parts of my Zettelkasten, I use “links” (i.e., I write in red ink the parenthesized ID of the referenced slip). This is the mechanical description of constructing links. But when would I want to do this?

Well, if we think of the Zettelkasten as a brain, and each slip as a neuron, a link is when two neurons are connected.

Now, I’m a mathematician, this may be overkill: do I really need to backlink to “Real Numbers” everytime I say something like “Let x be a real number”? This is a judgment call. For something as elementary as “real number” (which is ingrained in us since elementary school), I would say no.

What about “vector space”? “Basis”? “Spanning set”?

Well, I would say, “It depends.”

If this were in the category of linear algebra, I would say, “No, because it’s the subject of the thread, or very near adjacent to it.”

On the other hand, in my subcategory of representation theory of Lie algebras, I tend to err on the side of back-linking.

I discussed some examples where links produced meaningful results in Brain as metaphor. My experience has taught me that “obvious links” have value, because following the links leads to surprises (useful insights).

I discussed on the referenced page three examples, which I quickly summarize as:

  1. I took notes on “time in economics”, adding Time (economics), Logical time, Mechanical time, and Historical time to my Zettelkasten. Here “historical time” refers to the entropic arrow of time familiar in physics, so in the contents of Historical time when I discuss “entropic arrow of time”, I link both to Entropy and arrow of time.

    At the same time, taking notes on Brouwer’s First act Intuitionism I need to discuss Brouwer’s interpretation of “a priori Intuition of time”. Ah, which “time” are we talking about? My Zettelkasten is familiar with at least three distinct notions of time in economics. I link to time (Economics) which then traces to Entropy and the arrow of time. Connecting Entropy with Brouwer’s Intuitionism is meaningful, useful, novel — hence surprising.

  2. I am writing about Dan Ingalls’s paper “Design Principles of Smalltalk” which treats Smalltalk as the medium of communication for a dialogue between the programmer and the computer. This leads to slips like Smalltalk (programming language), Dialogue with computer (which links to Dialogue, and I then update Dialogue to “backlink” to Dialogue with computer), and Computer has body and mind. However, Dialogue links to Communication, which links to Autopoiesis. This leads to the intriguing connection between programming and autopoiesis, which I have not seen anyone discuss.

  3. Extending the previous example, a proof assistant facilitates dialogue between the user and computer, concerning formalizing Mathematics within a foundations of Mathematics. You literally engage in a Dialogue with computer in the manner discussed in the design principles of Smalltalk.

    Moreover, Brouwer thought this dialogue was a mental activity, which has been interpreted as suggesting Intuitionism was a solipsistic view of Mathematics. But thinking about proof assistant as a participant in the dialogue gives a fresh perspective on Intuitionism. One I have not seen adequately discussed, except perhaps Paul Lorenzen’s work.

By “wandering” the links, we end up with meaningful insights which are surprising. Each of the links are “obvious” (Dialogue links with Dialogue with computer and vice-versa, Intuitionism discusses “a priori Intuition of time” and so it links with Time, and so on). Nothing is surprising when making the links. Chasing the links produces insights.

That’s the “serendipity” of the Zettelkasten. Consequently, the “essence” of Luhmann’s Zettelkasten can boil down to two essential ingredients:

  1. Permanent ID numbers for notes (where each note discusses one idea), and
  2. Links/references using those permanent ID numbers.

It takes some experiment (trial-and-error) to figure out “how much” writing constitutes “one idea”.

Generalization of Existing Concepts

I wrote my Zettelkasten as if it were an intelligent-but-ignorant undergraduate who didn’t take much math in secondary school. Consequently, I introduced linear algebra in multi-staged, cyclical manner: elementary linear algebra tries to solve systems of equations. When I wrote it, my colleague only knew elementary algebra (there are numbers, variables, and functions…oh, and we pretend there is a “number” called “i” which is the square-root of negative one). When I defined a vector space in this setting, it was restricted implicitly to be over the real numbers.

Later, in my discussion of intermediate linear algebra, this would roughly correspond to the “first genuine undergraduate math course”. I introduced a notion of a “field of numbers”, and then a “vector space over a field”. In a remark to the definition of a vector space over a field, I remarked how it generalizes our earlier notion (linked back to the earlier notion), and discussed how.

Then, I went back to the earlier definition of a vector space, and at the bottom write in parentheses “(This is generalized to other number systems later <link>”. I did not close the parentheses, because I had the foresight to realize I would generalize it further to “modules over rings”, but it would be entirely valid to have closed the parentheses.