Applied Category Theory (ACT) as “precision machining”

Last week, while attending NIST’s 2022 workshop on compositional modeling, ably led by Priyaa and Joe – thanks to both! – I claimed that “precision machining is a good metaphor for applied category theory”, for which I would like to fill in some details.

Let’s discuss.

Idea 1: So far as I know, there are three main uses for precision maching:

1. for tool and die-making, where the results will be used to make millions of cheap interoperable parts – think molds for injection molding
2. for making “singular objects”, where a object or assembly needs to work or to fit “perfectly” – think the James Webb Space Telescope
3. for prototyping, where the meta-goal is to optimize the prototyping process itself around flexibility and minimal cycle time

Idea 2: I think these line up with the main uses, value-benefits, and pain-points I see applied category theory addressing:

1. ACT can be used for a kind of up-leveled “domain driven design”. Here, it plays both the role of a medium – the material out of which designs are shaped – and of the shaping tools – for precisely modeling phenomena of interest – what David Spivak calls “crystallizing” a “system of accounting” for the phenomena to be modeled.

2. ACT can be used for precision fits – e.g., when one needs to prove that a particular design, object, or family of objects has some essential property, or when one needs to study precisely in which ways a given piece falls short. In this sense, ACT is providing a source of “gauges” for testing fit as well as a, potentially, a “coordinate measuring machine” for quickly establishing the vital statistics of the object being built.

3. ACT can be used for prototyping – as a weird kind of programming language, ACT plays roles a bit like those played by specialized languages like Forth, Datalog, and Alloy – not necessarily what you’d implement your production version in, but in proficient hands, a fantastically powerful tool for narrowing in on “what is needed”.

Idea 3: The histories will rhyme.

1. There will be a success that solves an important but narrow problem that unlocks commercial possiblities. In machining, this may have been Maudsley’s screw-cutting lathe. In ACT, this may be ologs?

2. There will be internal informational developments in the field that completely change the economics and ergonomics of its application by practitioners. In machining, these developments included the codification and distribution of knowledge that enabled interoperability between practitioners at the level of both commiunication and of division of labor – things like the development of shared conventions for mechanical drawings, the codification of metrology and metallurgy, as well as the availability of “shop tooling” like vises, gauge blocks, and interchangeable tooling.

3. There will be breakthrough products and companies in the inner field that will enable practitioners to work much more efficiently and ergonomically, and that will wildly alter the field’s economics. In machining, think of Bridgeport Machines or the Moore Special Tools Company. (Note: I’m not saying that consumers will ever come near these specialized tools, but instead that the effect of the tools built with them may be much more widely felt.)

Restatement

In one way, above, I’m mostly just claiming that:

• ACT is math,
• math is hard to shape but can be made very precise at need, and
• ACT, as a species of math, is good for making models of some parts of the world I think are important

But in another way, I’m claiming something more: of all the kinds of math we (and I) know so far, ACT is uniquely exciting to me as a kind of material (and a box of tools) to model parts of the world with… because I find it convenient and promising in way that seems different from everything else.

Background

I’m a student of category theory for one of the same reasons that I’m a huge fan of Donald Schön’s concept of generative metaphors: I see both bodies of work as being among the best tools we have for understanding and wielding the “seeing as” process to help move ourselves through the world.

Further reading

Some resources for learning more about ACT include:

• Bill Lawvere & Stephen Schanuel’s “Conceptual Mathematics” (publisher, archive.org)
• Brendan Fong & David Spivak’s “Seven Sketches of Compositionality” (publisher, open access pdf, course website)
• John Baez’ & others talks and writings on “decorated cospans”, and similar technologies for describing “open networks”
• Pawel Sobocinski’s “Graphical Linear Algebra” site
• The Topos Institute blog and events

Some eclectic resources for learning more about precision machining include, (leaving aside important but high-commitment formal educational offerings such as those from trade schools or graduate mechanical engineering programs):

• The Moore books:

  • Holes, contours, and surfaces
  • Precision hole location
  • Foundations of mechanical accuracy

• The Design of Machine Elements

• Machinery’s Handbook

An extensive library of videos on YouTube documenting projects; “how is it made?” videos; shop, lab, and factory tours; and tool and instrument manufacturer processes and offerings, including entertaining favorites such as:

• Blondihacks, educational material for hobbyists
• This Old Tony, humor,
• Applied Science, garage recreations of commercially important processes
• Robin Renzetti, on precision spindles, instruments, tooling, and support equipment (precision ground stones)
• Dan Gelbart, on tools for prototyping, and an inspiring air-bearing lathe