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Loading a second brain

Before ChatGPT, the hype among productivity boosters was a PKMs or Personal knowledge management system.

It gained popularity through Tiago Forte’s book ‘Building a second brain’, and (for academics perhaps a more useful read) ‘How to take smart notes’ by Sönke Ahrens.



These books promote new techniques for note-taking (and for storing these notes) such as the PARA-method, the CODE-system, and Zettelkasten.

Unmistakable Creative has some posts on the principles behing the ‘second brain’ approach.

Your brain isn’t like a hard drive or a dropbox, where information is stored in folders and subfolders. None of our thoughts or ideas exist in isolation. Information is organized in a series of non-linear associative networks in the brain.

Networked thinking is not just a more efficient way to organize information. It frees your brain to do what it does best: Imagine, invent, innovate, and create. The less you have to remember where information is, the more you can use it to summarize that information and turn knowledge into action.

and

A network has no “correct” orientation and thus no bottom and no top. Each individual, or “node,” in a network functions autonomously, negotiating its own relationships and coalescing into groups. Examples of networks include a flock of birds, the World Wide Web, and the social ties in a neighborhood. Networks are inherently “bottom-up” in that the structure emerges organically from small interactions without direction from a central authority.

-Tiago Forte, Tagging for Personal Knowledge Management

There are several apps you can use to start building your second brain, the more popular seem to be Roam Research, LogSeq, and Obsidian.

These systems allow you to store, link and manipulate a large collection of notes, query them as a database, modify them in various ways via plugins or scripts, and navigate the network created via graph-views.

Exactly the kind of things we need to modify the simple system from the shape of languages-post into a proper topos of the unconscious.

I’ve been playing around with Obsidian which I like because it has good LaTeX plugins, powerful database tools via the Dataview plugin, and one can execute codeblocks within notes in almost any programming language (python, haskell, lean, Mathematica, ruby, javascript, R, …).

Most of all it has a vibrant community of users, an excellent forum, and a well-documented Obsidian hub.

There’s just one problem, I’m a terrible note-taker, so how can I begin to load my ‘second brain’?

Obsidian has several plugins to import data, such as your Kindle highlights, your Twitter feed, your Readwise-data, and many others, but having been too lazy in the past, I cannot use any of them.

In fact, the only useful collection of notes I have are my blog-posts. So, I’ve uploaded NeverEndingBooks into Obsidian, one note per post (admittedly, not very Zettelkasten-like), half a million words in total.

Fortunately, I did tag most of these posts at the time. Together with other meta-data this results in the Graph view below (under ‘Files’ toggled tags, under ‘Groups’ three tag-colours, and under ‘Display’ toggled arrows). One can add colour-groups based on tags or other information (here, red dots are posts tagged ‘Grothendieck’, the blue ones are tagged ‘Conway’, the purple ones tagged ‘Connes’, just for the sake of illustration). In Obsidian you can zoom into this graph, place a pointer on a node to highlight the connecting dots, and much more.



Because I tend to forget such things, and as it may be useful to other people running a WordPress-blog making heavy use of MathJax, here’s the procedure I followed:

1. Follow the instructions from Convert wordpress articles to markdown.

In the wizard I’ve opted to go only for yearly folders, to prefix posts with the date, and to save all images.

2. This gives you a directory with one folder per year containing markdown versions of your posts, and in each year-folder a subfolder ‘img’ containing all images.

Turn this directory into an Obsidian-vault by opening Obsidian, click on the ‘open another vault’ icon (third from bottom-left), select ‘Open folder as vault’ and navigate to your directory.

3. You will notice that most of your LaTeX cannot be parsed because during the markdown-process backslashes are treaded as special character, resulting in two backslashes for every LaTeX-command…

A remark before trying to solve this: another option might be to use the wordpress-to-hugo-exporter, resulting in clean LaTeX, but lacking the possibility to opt for yearly-folders (it dumps all posts into one folder), and it makes a mess of the image-files.

4. So, we will need to do a lot of search&replaces in all files, and need a convenient tool for this.

First option was the Sublime Text app, which is free and does the search&replaces quickly. The problem is that you have to save each of the files, one at a time! This may take hours.

I’ve done it using the Search and Replace app (3$) which allows you to make several searches/replaces at the same time (I messed up LaTeX code in previous exports, so needed to do many more changes). It warns you that it is dangerous to replace strings in all files (which is the reason why Sublime Text makes it difficult), you can ignore it, but only after you put the ‘img’ folders away in a safe place. Otherwise it will also try to make the changes to these files, recognise that they are not text-files, and drop them altogether…

That’s it.

I now have a backup network-version of this blog.



As we mentioned in the previous post a first attempt to construct the ‘topos of the unconscious’ might be to start with a collection of notes (the ‘conscious’) and work on the semantics of text-snippets to unravel (a part of) the unconscious underpinning of these notes. We also mentioned that the poset-structure in that post should be replaced by a more involved network structure.

What interests me most is whether such an approach might be doable ‘in practice’, and Obsidian looks like the perfect tool to try this out.

What we need is a sufficiently large set of notes, of independent interest, to inject into Obsidian. The more meta it is, the better…

(tbc)

Previously in this series:

Next:

The enriched vault

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the future of this blog (2)

is decided : I’ll keep maintaining this URL until new-year’s eve. At that time I’ll be blogging here for 5 years…

The few encounters I’ve had with architects, taught me this basic lesson of life : the main function of several rooms in a house changes every 5 years (due to children and yourself getting older).

So, from january 1st 2009, I’ll be moving out of here. I will leave the neverendingbooks-site intact for some time to come, so there is no need for you to start archiving it en masse, yet.

Previously I promised to reconsider this blog’s future over a short vacation, but as vacation is looking to be as illusory as the 24-dimensional monster-manifold, I spend my time throwing up ideas into thin and, it seems, extremely virtual air.

Some of you will think this is a gimmick, aiming to attract more comments (there is no post getting more responses than an imminent-end-to-this-blog-post) but then I hope to have settled this already. Neverendingbooks will die on 31st of december 2008. The only remaining issue being : do I keep on blogging or do I look for another time-consumer such as growing tomatoes or, more probably, collecting single malts…

For reasons I’ve stated before, I can see little future in anything but a conceptual-, group- blog. The first part I can deal with, but for the second I’ll be relying on others. So, all I can do is offer formats hoping that some of you are willing to take the jump and try it out together.

Such as in the bloomsday-post where I sketched the BistroMath blog-concept. Perhaps you thought I was just kidding, hoping for people to commit themselves and them calling “Gotcha…”. Believe me, 30 years of doing mathematics have hardwired my brains such that I always genuinely believe in the things I write down at the moment I do (but equally, if someone offers me enough evidence to the contrary, I’ll drop any idea on the spot).

I still think the BistroMath-project has the potential of leading to a bestseller but Ive stated I was not going to pursue the idea if not at least 5 people were willing to join and at least 1 publisher showed an interest. Ironically, I got 2 publishers interested but NO contributors… End of that idea.

Today I offer another conceptual group-blog : the Noether-boys seminar (with tagline ; _the noncommutative experts’ view on 21st century mathematics_). And to make it a bit more concrete Ive even designed a potential home-page :



So, what’s the deal? In the 1930-ties Emmy Noether collected around her in Goettingen an exceptionally strong group of students and collaborators (among them : Deuring, Fitting, Levitski, Schilling, Tsen, Weber, Witt, VanderWaerden, Brauer, Artin, Hasse, MacLane, Bernays, Tausky, Alexandrov… to name a few).

Collectively, they were know as the “Noether-boys” (or “Noether-Knaben” or “Trabanten” in German) and combined seminar with a hike to the nearby hills or late-night-overs at Emmy’s apartment. (Btw. there’s nothing sexist about Noether-boys. When she had to leave Germany for Bryn Mawr College, she replaced her boys to form a group of Noether-girls, and even in Goettingen there were several women in the crowd).

They were the first generation of mathematicians going noncommutative and had to struggle a bit to get their ideas accepted.
I’d like to know what they might think about the current state of mathematics in which noncommutativity seems to be generally accepted, even demanded if you want to act fashionable.

I’m certain half of the time they would curse intensely, and utter something like ‘steht shon alles bei Frau Noether…’ (as Witt is witnessed to have done at least once), and about half the time they might get genuinely interested, and be willing to try and explain the events leading up to this to their fellow “Trabanten”. Either way, it would provide excellent blog-posts.

So I’m looking for people willing to borrow the identity of one of the Noether-boys or -girls. That is, you have to be somewhat related to their research and history to offer a plausible reaction to recent results in either noncommutative algebra, noncommutative geometry or physics. Assuming their identity you will then blog to express your (that is, ‘their’) opinion and interact with your fellow Trabanten as might have been the case in the old days…

I’d like to keep Emmy Noether for the admin-role of the blog but all other characters are free at this moment (except I’m hoping that no-one will choose my favourite role, which is probably the least expected of them anyway).

So please, if you think this concept might lead to interesting blogging, contact me! If I don’t get any positives in this case either, I might think about yet another concept (or instead may give up entirely).

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Arnold’s trinities

Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic…

David Corfield of the n-category cafe and a dialogue on infinity (and perhaps other blogs I’m unaware of) pointed me to the paper Symplectization, complexification and mathematical trinities by Vladimir I. Arnold. (Update : here is a PDF-conversion of the paper)

The paper is a write-up of the second in a series of three lectures Arnold gave in june 1997 at the meeting in the Fields Institute dedicated to his 60th birthday. The goal of that lecture was to explain some mathematical dreams he had.

The next dream I want to present is an even more fantastic set of theorems and conjectures. Here I also have no theory and actually the ideas form a kind of religion rather than mathematics.
The key observation is that in mathematics one encounters many trinities. I shall present a list of examples. The main dream (or conjecture) is that all these trinities are united by some rectangular “commutative diagrams”.
I mean the existence of some “functorial” constructions connecting different trinities. The knowledge of the existence of these diagrams provides some new conjectures which might turn to be true theorems.

Follows a list of 12 trinities, many taken from Arnold’s field of expertise being differential geometry. I’ll restrict to the more algebraically inclined ones.

1 : “The first trinity everyone knows is”

where $\mathbb{H} $ are the Hamiltonian quaternions. The trinity on the left may be natural to differential geometers who see real and complex and hyper-Kaehler manifolds as distinct but related beasts, but I’m willing to bet that most algebraists would settle for the trinity on the right where $\mathbb{O} $ are the octonions.

2 : The next trinity is that of the exceptional Lie algebras E6, E7 and E8.

with corresponding Dynkin-Coxeter diagrams

Arnold has this to say about the apparent ubiquity of Dynkin diagrams in mathematics.

Manin told me once that the reason why we always encounter this list in many different mathematical classifications is its presence in the hardware of our brain (which is thus unable to discover a more complicated scheme).
I still hope there exists a better reason that once should be discovered.

Amen to that. I’m quite hopeful human evolution will overcome the limitations of Manin’s brain…

3 : Next comes the Platonic trinity of the tetrahedron, cube and dodecahedron



Clearly one can argue against this trinity as follows : a tetrahedron is a bunch of triangles such that there are exactly 3 of them meeting in each vertex, a cube is a bunch of squares, again 3 meeting in every vertex, a dodecahedron is a bunch of pentagons 3 meeting in every vertex… and we can continue the pattern. What should be a bunch a hexagons such that in each vertex exactly 3 of them meet? Well, only one possibility : it must be the hexagonal tiling (on the left below). And in normal Euclidian space we cannot have a bunch of septagons such that three of them meet in every vertex, but in hyperbolic geometry this is still possible and leads to the Klein quartic (on the right). Check out this wonderful post by John Baez for more on this.



4 : The trinity of the rotation symmetry groups of the three Platonics

where $A_n $ is the alternating group on n letters and $S_n $ is the symmetric group.

Clearly, any rotation of a Platonic solid takes vertices to vertices, edges to edges and faces to faces. For the tetrahedron we can easily see the 4 of the group $A_4 $, say the 4 vertices. But what is the 4 of $S_4 $ in the case of a cube? Well, a cube has 4 body-diagonals and they are permuted under the rotational symmetries. The most difficult case is to see the $5 $ of $A_5 $ in the dodecahedron. Well, here’s the solution to this riddle



there are exactly 5 inscribed cubes in a dodecahedron and they are permuted by the rotations in the same way as $A_5 $.

7 : The seventh trinity involves complex polynomials in one variable

the Laurant polynomials and the modular polynomials (that is, rational functions with three poles at 0,1 and $\infty $.

8 : The eight one is another beauty

Here ‘numbers’ are the ordinary complex numbers $\mathbb{C} $, the ‘trigonometric numbers’ are the quantum version of those (aka q-numbers) which is a one-parameter deformation and finally, the ‘elliptic numbers’ are a two-dimensional deformation. If you ever encountered a Sklyanin algebra this will sound familiar.

This trinity is based on a paper of Turaev and Frenkel and I must come back to it some time…

The paper has some other nice trinities (such as those among Whitney, Chern and Pontryagin classes) but as I cannot add anything sensible to it, let us include a few more algebraic trinities. The first one attributed by Arnold to John McKay

13 : A trinity parallel to the exceptional Lie algebra one is

between the 27 straight lines on a cubic surface, the 28 bitangents on a quartic plane curve and the 120 tritangent planes of a canonic sextic curve of genus 4.

14 : The exceptional Galois groups

explained last time.

15 : The associated curves with these groups as symmetry groups (as in the previous post)

where the ? refers to the mysterious genus 70 curve. I’ll check with one of the authors whether there is still an embargo on the content of this paper and if not come back to it in full detail.

16 : The three generations of sporadic groups

Do you have other trinities you’d like to worship?

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