After mulling over my bullet points, it occurred to me that the network problems I was dealing with—background cloud sync, editing across multiple devices, real-time collaboration, offline support, and reconciliation of distant or conflicting revisions—were all pointing to the same question: was it possible to design a system where any two revisions of the same document could be merged deterministically and sensibly without requiring user intervention?

It’s what happened after sync that was troubling. On encountering a merge conflict, you’d be thrown into a busy conversation between the network, model, persistence, and UI layers just to get back into a consistent state. The data couldn’t be left alone to live its peaceful, functional life: every concurrent edit immediately became a cross-architectural matter.

I kept several questions in mind while doing my analysis. Could a given technique be generalized to arbitrary and novel data types? Did the technique pass the PhD Test? And was it possible to use the technique in an architecture with smart clients and dumb servers?

Concurrent edits are sibling branches. Subtrees are runs of characters. By the nature of reverse timestamp+UUID sort, sibling subtrees are sorted in the order of their head operations.

This is the underlying premise of the Causal Tree. In contrast to all the other CRDTs I’d been looking into, the design presented in Victor Grishchenko’s brilliant paper was simultaneously clean, performant, and consequential. Instead of dense layers of theory and labyrinthine data structures, everything was centered around the idea of atomic, immutable, metadata-tagged, and causally-linked operations, stored in low-level data structures and directly usable as the data they represented.

I’m going to be calling this new breed of CRDTs operational replicated data types—partly to avoid confusion with the exiting term “operation-based CRDTs” (or CmRDTs), and partly because “replicated data type” (RDT) seems to be gaining popularity over “CRDT” and the term can be expanded to “ORDT” without impinging on any existing terminology.

Much like Causal Trees, ORDTs are assembled out of atomic, immutable, uniquely-identified and timestamped “operations” which are arranged in a basic container structure. (For clarity, I’m going to be referring to this container as the structured log of the ORDT.) Each operation represents an atomic change to the data while simultaneously functioning as the unit of data resultant from that action. This crucial event–data duality means that an ORDT can be understood as either a conventional data structure in which each unit of data has been augmented with event metadata; or alternatively, as an event log of atomic actions ordered to resemble its output data structure for ease of execution

To implement a custom data type as a CT, you first have to “atomize” it, or decompose it into a set of basic operations, then figure out how to link those operations such that a mostly linear traversal of the CT will produce your output data. (In other words, make the structure analogous to a one- or two-pass parsable format.)

OT and CRDT papers often cite 50ms as the threshold at which people start to notice latency in their text editors. Therefore, any code we might want to run on a CT—including merge, initialization, and serialization/deserialization—has to fall within this range. Except for trivial cases, this precludes O(n2) or slower complexity: a 10,000 word article at 0.01ms per character would take 7 hours to process! The essential CT functions have to be O(nlogn) at the very worst.

Of course, CRDTs aren’t without their difficulties. For instance, a CRDT-based document will always be “live”, even when offline. If a user inadvertently revises the same CRDT-based document on two offline devices, they won’t see the familiar pick-a-revision dialog on reconnection: both documents will happily merge and retain any duplicate changes. (With ORDTs, this can be fixed after the fact by filtering changes by device, but the user will still have to learn to treat their documents with a bit more caution.) In fully decentralized contexts, malicious users will have a lot of power to irrevocably screw up the data without any possibility of a rollback, and encryption schemes, permission models, and custom protocols may have to be deployed to guard against this. In terms of performance and storage, CRDTs contain a lot of metadata and require smart and performant peers, whereas centralized architectures are inherently more resource-efficient and only demand the bare minimum of their clients. You’d be hard-pressed to use CRDTs in data-heavy scenarios such as screen sharing or video editing. You also won’t necessarily be able to layer them on top of existing infrastructure without significant refactoring.

Perhaps a CRDT-based text editor will never quite be as fast or as bandwidth-efficient as Google Docs, for such is the power of centralization. But in exchange for a totally decentralized computing future? A world full of devices that control their own data and freely collaborate with one another? Data-centric code that’s entirely free from network concerns? I’d say: it’s surely worth a shot!

If I had to guess why "math reform" is misinterpreted as "math education reform", I would speculate that school is the only contact that most people have had with math. Like school-physics or school-chemistry, math is seen as a subject that is taught, not a tool that is used. People don't actually use math-beyond-arithmetic in their lives, just like they don't use the inverse-square law or the periodic table.

Teach the current mathematical notation and methods any way you want -- they will still be unusable. They are unusable in the same way that any bad user interface is unusable -- they don't show users what they need to see, they don't match how users want to think, they don't show users what actions they can take.

They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions. Represent things visually and tangibly. And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface.

Anything that remains abstract (in the sense of not concrete) is hard to think about... I think that mathematicians are those who succeed in figuring out how to think concretely about things that are abstract, so that they aren't abstract anymore. And I believe that mathematical thinking encompasses the skill of learning to think of an abstract thing concretely, often using multiple representations – this is part of how to think about more things as "things". So rather than avoiding abstraction, I think it's important to absorb it, and concretize the abstract... One way to concretize something abstract might be to show an instance of it alongside something that is already concrete.

The mathematical modeling tools we employ at once extend and limit our ability to conceive the world. Limitations of mathematics are evident in the fact that the analytic geometry that provides the foundation for classical mechanics is insufficient for General Relativity. This should alert one to the possibility of other conceptual limits in the mathematics used by physicists.