The two parts of my study of mathematics are
- frontier-expansion with MathAcademy, and
- hyperobject-unification with some mathematical field (and then onward to the next one)
The former should be fairly self-explanatory — it means learning the basics of some field by doing the corresponding MathAcademy course. The objective of this part is to gain a basic understanding of the constituents of the hyperobject, and basic fluency with solving problems using them and their operations. Spaced repetition is used here, and this is taken care of by MathAcademy’s system.
The Elements of Hyperobject-unification
The latter part — getting to hyperobject-unification with some mathematical field (currently, for instance, classical geometry) has two and a half streams:
- Constructing the hyperobject as a structure of understanding, getting comfortable with it, and committing it to memory
- Developing virtuosity/spontaneity/deep fluency in its handling and manipulation and deep connectome
- For flavour and out of interest: getting into the context, history, meta, and lore of the field.
The first two both use spaced repetition in their own way.
Construction, basic fluency, deep understanding
The first stream is for constructing and getting comfortable with, familiar with, the hyperobject and its basic structure and manipulations. The objective of this phase is to a) construct the hyperobect as hyperobject, b) deeply understanding it and its parts, and c) understanding (and internalising!) its basic connectome.
(I assume that MathAcademy has already provided a foundational understanding of the parts, of what I call the skeletal connectome (ie, the dependency graph), and of solving problems with this foundational understanding.)
It means picking a field (right now, classical geometry, say), then strengthening what I learnt in MA (or wherever else) and building connections between the somewhat sparse constituents of the pedagogical dependency graph by first going through a few pedagogic textbooks, then going through a few aesthetic ones (an ‘aesthetic’ textbook is one which has a ’take’, a certain coherence, a (good!) ’lens’, a particular construction of or perspective on the hyperobject which informs the text, of which (in some sense) the text is in fact an emanation; it’s like knowing a thing you know but from a different angle, or seeing a particular ‘side’ of a person that you perhaps otherwise knew quite well), and finally a few ‘hardcore’ texts, which do little to no hand-holding.
Immersion, virtuoso deftness, intimate understanding
This second stream is for a) immersing in and comfortably traversing the hyperobject’s loka, its higher-dimensional reality, b) becoming deft at playing with it, with manipulating it, of tactile virtuosity with it, of holding it, as if it were a single complex things that you treated as such, and c) an intimacy with, a felt sense for, its rich connectome, for its deep patterns.
This looks like being comfortable not just with the constituent pieces/insights of the hyperobject, nor just the basic connections, but with the rich connectivity of its parts, with its extended connectome, with its repeating patterns; immersing onesels with its deep structure (if it has one). Another is becoming deft in its manipulation, tactile; developing a sense for it; seeing through it, in a sense. The goal is getting to the point where what may eventually be legibilised as a complex multistep proof or solution to a problem feels internally like a straight-line traversal through the hyperobject’s higher-dimensional reality/structure.
In practice: doing lots and lots and lots of problems, increasing in difficulty, intricacy, and combinatorial (or in chess terms, ‘combinational’) complexity (ie, where an entrainment of more than just one piece of the hyperobject is required to solve the problem, while also requiring natural fluency with the way the pieces/chunks of the whole connect with each other).
Flavour and interest
The third stream — half-stream — is getting into the field’s context, meta, history, and lore. Perhaps skip if not motivated by interest or curiosity or an outwarding openness or (maybe) obsession.
The Role of (sometimes Spaced) Repetition
For MathAcademy, as noted above, nothing needs be done by the student WRT spaced repetition. Nor for the half-stream, in my opinion.
For hyperobject construction (first stream)
For the first stream, this kicks in only at the level of working through some ‘hardcore’ text that would qualify as a root-text. Here, the mode of study includes something like Nielsen’s strategy in using spaced repetition to see through a piece of mathematics.
It is here that you ‘SRify’ the material (like I am doing with Go, and intend to do with a bunch of others) and commit the part(s) of it suitable to such treatment to memory that way. (At this point, having worked through both pedagogic and aesthetic texts, I assume that you’re not likely to fall into the more beginner classes of SRification errors when doing this.)
For proofs, the way is as given in the above link, combined with adding any proofs you find difficult to the deep practice book (linked below). To be clear, I don’t mean just proofs as practice, I mean proofs of the fundamental results of the field as well. The idea is to have the field and its structure completely internalised. Treating the basic results as ‘special’ and only proofs assigned as exercises or problems as fit for inclusion in the deep practice book is the opposite of this. You want the heart and soul of the field, not just its applications, to be yours.
Since I haven’t got to this phase yet, I’m not sure whether it’s a good idea to have separate problem and proof sections, or treat proofs of fundamental theroems as merely another ‘problem’, whose solution merely happens to be an argument that’s the proof of a proposition. I lean towards the latter; away with enervating artificial distinctions!
For Hyperobject immersion (second stream)
The strategy here is keeping a Deep Practice book, starting as instructed by Hendrickson, while working through increasingly difficult problems and problem books.
There’s an element of deliberate practice here, in that the parts you are weakest at are the ones most likely to end up here, which means you’ll end up practising those more.