Learning to Learn Again

Learning
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I find myself remembering a time when I could sit down for hours and hours poring over a textbook and working through exercises, days when I could devour a novel from cover to cover in one very long sitting, and chunks of months when I could start a course and finish it (but to be fair, online courses are much more difficult to do that for). Life has taken me and my cognitive ability to a place where that kind of learning is no longer possible. Believe me, I’ve tried repeatedly. It’s just too costly to keep trying to go for linear learning.

One reason is life — my context and my brain are no longer what they were all those years ago. The other reason is that the subjects that have captured my heart are difficult to access. What I mean by that is that there’s no online course, no standard textbook, no learning material on category theory and active inference that I have yet found that fulfils these two criteria I have for learning:

  1. start where I am, and
  2. keep me focused.

This doesn’t mean that there aren’t great things out there — lots are written by many who write here and are in the wider ecosystem of the Topos Institute. In fact, it was David and Brendan who invited me into the world of category theory, via an Amazon review search mid-pandemic, but that’s another story for another post. The problem is that all of the resources when I started studying were for working mathematicians, scientists, programmers… I’m none of the above. So I did give them a good go, but had to stop almost all of them. I’m grateful for David and Brendan for “leaving open the opportunity for [me] to come back to the book at a later date and get more deeply into it” (Fong & Spivak p. v), and I think of every textbook I’ve had to stop going through that way. When I grow up, I’ll be able to go through them.

The point is that self-learners of category theory are doing it! And have been for decades, from that wonerful yellow and cream Mac Lane book! So I must be able to do it somehow. And thanks to words by authors like Eugenia Cheng (The Joy of Abstraction I did manage to read cover to cover and participate in the Book Club), David, and Brendan, to watching the Topos Institute Outreach Panel, and to the active and encouraging education stream of Active Inference Institute open science community, I’ve been able to find and grow into a way forward for myself. And that way is the adventure learning way: jump right in, hold on to your butt, and see where you get to.

Adventure syllabi

I’m sure a ton of people are already doing this; the thing is that I’d rather learn about the thing itself than to learn about learning a thing. I don’t have the time or cognitive capacity to learn about learning through a course. This adventure way also goes against a lot of the conventional wisdom for learning. I don’t have a schedule, I don’t have things in my calendar. I can only work towards stuff in snatches of time.

Well, in any case, here is my prototype for adventure learning (and my running example):

  1. Take a really hard thing. Take another really hard thing. So you have hard thing x hard thing. (category theory x active inference)
  2. Find one print thing that covers both of these things (Toby’s dissertation - I adventured in version 1 of it)
  3. Adventure through it - individually or with a group (I went through it with Active Inference Category Theory learning group 2023; there was not a category theoretician among us)

By “adventure through it”, I mean explore and battle through while creating at least one artefact. I jump around. I am also allowed to and must jump around all the other category theory books and websites from Wikipedia to nLab to nLab cafe are at my disposal and that’s not only okay - I’ve found it necessary to generate many and varied observations.

The artefact is important — it’s the thing that ensures criteria 2 above, keep me focused, is fulfilled. It doesn’t have to be an artifact of perfect quality — in fact, it must be suboptimal. That way, it can continue to be worked on as I learn.

In my example, I have two artefacts:

  1. A story blog at https://jrandthesecretcitadel.wordpress.com/, and
  2. An adventure syllabus for exploring the intersection between category theory and active inference centered around Toby’s dissertation and Eugenia’s book — for those who prefer to (or must, like me) jump right in and jump around to learn in a non-linear kind of way (on Github GitHub - ActiveInferenceInstitute/ActiveInferenceCategoryTheory: Active Inference & Category Theory)

My first criteria was also fulfilled in this, because the dissertation (math papers in general) was so completely alien to me that I had almost no priors coming in. (I’m not a biologist, I’m not a neurscientist!). So it made me feel challenged, completely disoriented, and yet… and yet — he explained things in a way that things I’d broken my brain on in MacLane seemed to make sense, at least at the level of intuition. So I was very pleased. In fact, I’ve read a lot of MacLane now — I’ve just jumped around. Toby’s dissertation was my anchor, and I consulted MacLand (and Riehl), to try to get a better grasp of stuff.

But how do I know I’ve learned?

The thing is, I don’t know know. At this stage, I have a feeling. I basically compare myself reading papers in January 2023 and myself reading papers in March 2024, and if I can read David Jaz Myers tell me to take a deep breath while he introduces the definition of a monad (p. 48 of his draft Categorical Systems Theory book), I don’t spontaneously poof turn into an elven paladin. I have a clue and I’m not thrown completely off the earth.

But in general, I’ve found that if I do think I understand something, it’s not because I answered a quiz question and check the answer key and got it right. I think I understand something when it just makes sense. When I can go through and follow, even if by the tippy-top-part of my brain, the logic of a proof, or the logic of the intiutive explanation.

Being able to correctly (sensibly, logically) apply the category theory I’ve learned (or think I’ve learned) to something will be the next step to knowing whether I’ve learned or not.

How do you talk to a Julia?

“Heaven and earth are moving / in my soul / I don’t know where to start” - Jamie Walters, How do you talk to an angel?

A few months ago, Toby recommended that I go through Aluffi’s Chapter 0 book alongside my first year undergrad math studies, and I’m just now buckling down to do that. This one may be the textbook I can get through, now that I’ve adventured in all sorts of different category theory places last year. So I’m giving my “previous/first/old” way of learning another go, for category theory at least.

However, my adventure learning has not ended. I don’t know how to code; again I’ve stop-start tried. However now, I’ve got another zany adventure. I am learning Julia by simultaneously going through RxInfer.jl active inference examples with the Active Inference Institute RxInfer.jl learning group, and by going through Doggo’s Julia for Absolute Beginners Full Course. Eventually, I’d like to then start going through Owen and Evan’s AlgebraicJulia tutorial (which I’ve started watching, but have paused on the installing bit). When I grow up!

I don’t quite have an artefact for this specifically yet, but I suspect that the artefact will be this learning framework I’m working on for learning active inference from any background. If I can apply category theory to this learning framework, like functorial semantics style, I’ll be a very happy elven paladin indeed.

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Awesome post! As someone who spends a lot of time doing self-guided learning, I found your post quite interesting. I like your analogies of “going on an adventure” and “creating an artifact”!

In response, I wanted to talk a bit about some things I’ve found to be helpful for self-learning math, in case this can be useful to you or someone else reading this. When I first started trying to self-learn math, I focused a lot on one book at a time, worked a lot of exercises, and tried to understand each section in detail before moving to the next one. This was helpful and I learned a lot, but nowadays I find that approach to be quite demanding - it takes a lot of energy.

The last few years, I’ve had a lot of fatigue (probably at least in part because I suffer from inflammatory bowel disease). So I’ve had to explore new ways to keep learning, trying to find ways that are compatible with a highly unpredictable and limited energy budget.

Here are some things that have worked for me:

  • I’ve found it helpful to collect a variety of resources on a variety of topics, at a variety of levels. Then, I try to listen to how I’m feeling on a given day. I try to pick a resource to learn from that sounds both interesting and not too difficult. There are two pitfalls here, at least for me: (1) focusing on “the basics” too much, and getting bored or (2) relentlessly focusing on material that is really hard for me, and burning out.
  • When I need something easier, I find it more manageable to work from a book with a lot of solutions in the back, or at least many detailed proofs.
  • I’ve recently begun to appreciate the importance of noticing when my brain needs a break, and then giving it one. Setting aside a very interesting exercise that I’ve only partially solved doesn’t have to be a bad thing - it can provide a great resumption point for when that topic sounds fun and doable again.
  • At least once a week, I like to learn about something completely different from what I’ve been learning about all week. Sometimes this can take the form of “mathematical tourism”, where I pick a book that’s too hard for me to understand in detail - and I just enjoy getting a very rough sense of some of the exciting ideas introduced there. This helps me remember how big math is, and how little I know about it - which is exciting. This can also help me realize when I’ve learned something - it’s a great feeling to discover I can now engage in some detail with a book that used to be largely opaque.
  • I find it helpful to keep a running analogy between abstract math concepts and applied topics of interest to me. I have a “scratchpad” file where I enjoy writing down some connections that occur to me, no matter how specific or vague. Sometimes I end up with certain perspectives or slogans that end up being helpful to my self-learning later.
  • I find it helpful to keep a list of questions in a file. There’s something about writing a question down that helps me think about it more clearly. I also find this very helpful for moving on from a question that is too difficult for me at the moment. If a question feels kind of important, sometimes I like to ask it on the category theory zulip (which is an amazing place!).

If I had to try and extract a key idea from my self-learning experience, maybe it would be this: I try to keep self-learning fun, exciting, and sustainable. Then even if my learning isn’t as “efficient” as it could be, at least I’ll enjoy it and probably keep doing it!

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Thanks, David! I think I fell into your Pitfall (2) and got quite lost there. These are really great points, and I’m encouraged by the overall feel of balance and fluidity that matches your brain capacity and energy level. LOL it sounds a lot like using The Force. (And thanks for the extra nudge into Zulip, I’ve finally joined!)

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Hey JR, thanks for posting, and your kind and interesting words! I owe you an e-mail; so thanks also for your patience there :slight_smile:

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Thanks Toby, no worries, and thank you! I know you’re busy, all in good time. Daniel just shared your protentions paper, so can’t wait to dig in.