U-lang, A-lang, and the Obs within

Oftentimes, I’ve found myself in strange territories on this lone wolf learning adventure. It’s familiar, yet dream-like in that the way that the familiar things (words, letters) are put together strike me as completely bizarre and alien. John Baez said it best in “Biology as Information Dynamics”, when he said, “I don’t know what to say about that… because I don’t… it doesn’t quite… I know what the words mean individually”. I don’t know what’s what. I don’t know what I’m looking at. I have no point of reference. The possible point of reference candidate may not only be unhelpful, but also actively get in the way. It’s like a rasterized mush of confusion — to me… but it’s the physical manifestation of an idea that obviously contains meaning to someone. The author wrote this, so there is something in their hidden state that they want to share with the world, and I want to be able to participate at this interface. I want to understand. But knowing that such understanding only comes with doing the work, at this point, I merely want to have a clue.

What I’ve found helpful in beginning to establish a point of reference is something I am learning from Haskell Curry in Foundations of Mathematical Logic (1977): U language, A language, and the Obs within them. Here are key quotations that form the basis of my current understanding:

U language

“The U language. Every investigation, including the present one, has to be communicated from one person to another by means of language. It is expedient to begin our study… by giving a name to the language being used, and by being explicit about a few of its features. … It is specific… contains technical terminology and other linguistic devices… generally understood by mathematicians of an appropriate degree of maturity… is continuously in process of growth… [and] by careful use we can attain any reasonable degree of precision.” (Curry 1977, p. 28-29)

A language

Professor Curry defines a “system”, in my understading, as a deductive theory containing “certain unspecified constitutents or parameters… that enter as unspecified objects about which the elementary statements assert that they have certain properties or that certain relations hold.” (Curry 1977, p. 50) He is very precise about his definition of “deductive theory” and “elementary statements” as well — I won’t go into that here. But the point is that we have to specify this language:

“In order to present such a system in the U language, it is necessary to decide on a notation for naming the formal objects and designating the basic predicates, and also on devices for combining these to form the U-sentences expressing the elementary statements. This notation, in its totality, forms a language in the semiotical sense; this language is here called the A language.” (Curry 1977, p. 50)

He goes on to say that A nouns are the names of formal objects (he also defines “formal objects” precisely), A verbs are basic predicates, and that A sentences are sentences that express elementary statements). But beware!

“It cannot, however, be too strongly emphasized that the A language is not a language being talked about; it is adjoined to the U language to be used therein. The A nouns are a special kind of U noun; the A verbs a special kind of U verb; and the A sentences a special kind of U sentence. The adjunction of this new terminology to the U language does not differ in any essential point from any other procedure where we introduce technical expressions into the U language.” (Curry 1977, p. 50-51)

Obs in A-lang

An ob is an element of a particular type of deductive system whose “formal objects form a monotectonic inductive class… Practically all the systems considered in modern mathematical logic and mathematics are of this character.” (Curry 1977, p. 54). He calls this system an ob system. Specifically, “The elements of this inductive class are called obs, its initial elements atoms, and its modes of… combination operations. Every ob is thus the result of a construction from the atoms by the primitive operations… it is irrelevant what the obs are; this colorless word ‘ob’ has been deliberately chosen to emphaisze this irrelevance.” (Curry 1977, p. 54)

Now as a side note, I understand that mathematical logic has evolved greatly since 1977! So I have no idea if these ideas have been updated, and what words folks use to talk about these things these days. There’s SO MUCH information out there and I am but a finite being. But this precious resource is available to me, and I’m using it. These three notions are proving to be helpful, and it largely has to do with my feelings.

Here is an example. With these ideas to hand, I can look at a page like this from Toby Smithe’s Mathematical Foundations for a Bayesian Brain (2023, v1, which was what I adventured through last year):

(St. Clere Smithe 2023v1, p. 26)

and I don’t run screaming into the night. I can say to myself, look! Here, Toby is speaking U-lang. This here, in this definition, this is A-lang and some Obs within the A-lang. So I get epistemic feelings — how I feel when I learn — of “I can stick with this” and proceed (often later in time) to look for a clue. (I got the idea of “epistemic feelings” from Fields and Glazebrook, who use cone-cocone diagrams, in ”Information flow in context-dependent hierarchical Bayesian inference” — also very helpful, but maybe that’s another post for another day.)

The clue is usually in the A-lang words you choose to use when you describe your formalism. It seems to work best when the reader already has prior knowledge of the branch of the math tree from which your A-lang term fruit has fallen out, such as topology. However, it’s helpful even to the likes of me, who does not have that prior knowledge in the mathematical context. Even that term from topology came from somewhere meaningful and observed by the person who came up with it. And that’s enough to get a finger- or toe-hold into that slippery clue-containing rock face. A-lang names for things are evocative. Examples are the A-lang terms “section” and “retraction”. Here’s an example which I encountered in Emily Riehl’s Category Theory in Context:

(Riehl 2016, p. 10)

The word “section” makes me imagine a slice of something; the word “retraction” makes me think of how a needle or ballpen retracts. These are valuable treasures on the quest for understanding.

Even your Obs afford me clues. When you choose a symbol to stand for values (a variable, OBviously), you’re already sharing information with me. For example, when you say you’re going to call your objects a, b, c, …, there’s an (alphabetical) order implicit in that choice. The same goes for your choice in calling morphisms f, g, h, …: by choosing a different section of the alphabet, you’re telling me you’re talking about a different kind of thing, and by using the same lower case notation, you’re telling me that you’re takling about something on the same level as objects. (Of course, there’s a “point of arbitrariness” that established the agreed-upon order of the latin alphabet. I highly recommend A Place for Everything: A Curious History of Alphabetical Order by Judith Flanders.) And when you decide to label natural transformations using Greek letters, like λ for the left unitor, and ρ for the right unitor, you’re telling me that you’re talking about another level altogether. Similar logic goes for using F, G, … for functors.

Finally, of course, your U-lang is what connects to me in the first instance. How you talk about your ideas in U-lang is where I find the sketch — the vision — of what you might be talking about. Most importantly, it’s usually the place where you tell me why. This helps because then I know where in my mental model world these new bits and pieces go — if not the exact place, at least in the general zone of my imagination the thing you’re about to tell me can possibly fit.

A recent example is from David Jaz Myers in his draft Categorical Systems Theory (forgive me, David, I’m jumping — I’m not actually at that page yet):

(Jaz Myers p. 254)

And one of my Great Learning Moments from Toby Smithe’s dissertation:

(St. Clere Smithe 2023v1, p. 100)

The greatest source of U-lang, however, is not actually on the page of your written work. It’s in your talks. I think William Thurston talked about this in “Proof and Progress in Mathematics”. For me, I’m talking about the videos available online. I wish I could attend talks in person, but I, as have tons of others, have been able to learn loads from watching recordings.

Of course, I’ve just been talking about clues and cues. I still don’t get get any of what I gave as examples above; the adventure continues. I think the point is that it’s the adventure itself that builds that longed-for understanding at the deeper level: the continuous updating of my prior knowledge, where the bigger the divergence, the bigger the learning opportunity, and the deeper the memory (especially when associated with epistemic feelings, positive and negative!).

That said, I will not say “no” to a beautiful explanation that speaks to me. One that fits my priors and that leads me to That Golden Update. Especially when I tap out on some of the most challenging concepts for me to understand understand (can you say, “degrees of freedom”?!). I continue my epistemic foraging for Legendary Explanations (bearing in mind that Legendary Explanations are very personal — one person’s Legendary Explanation may be another’s Mush of Confusion).

Without access to these territories and terrains full of travails and treasures, I’d have nothing to adventure through, no U- or A-lang to appeal to. So many thanks to you all who put PDFs and writings and recordings of your U-lang and A-lang territories full of treasures online in front of the paywall (and to all those who fund you and work for making that possible). You doing that makes this:

(Quoted in Riehl 2016, p. iii)

possible for the likes of me, who, for multitudes of reasons unique to each adventurer, can’t enroll in a recognized, credited class on category theory. #Gratitude

1 Like