Links - Zettelkasten

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.

Special Cases Calling for Links, Back-Links

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.