The object/meta distinction is sort of weird to maintain here; its weirdness has to do with us being reflective creatures, always thinking together about a domain and about us [thinking about]/[doing stuff in] the domain. When you are engaged in any challenging activity, you’re not solely “making very object-level moves”, but also thinking about the object-level moves you’re making; now, of course, the thinking about object-level moves you’re making can also itself naturally be seen as consisting of moves in that challenging activity, blurring the line between object-level-moves and meta-moves. And I’m just saying this is true in particular for challenging thinking-activities; in such activities, the object-moves are already more like thought-moves, and the meta-moves are/constitute/involve thinking about your thinking (though you needn’t be very explicitly aware that this is what you’re doing, and you often won’t be). A mathematician doesn’t just “print statements” without looking at the “printing process”, but is essentially always simultaneously seeking to improve the “printing process”. And this isn’t at all unique to mathematics — when a scientist or a philosopher asks a question, they also ask how to go about answering that question, seek to make the question make more sense, etc.. (However, it is probably moderately more common in math than in science for things which first show up in thinking to become objects of study; this adds to the weirdness of drawing a line between object and meta in math. Here’s one stack of examples: one can get from the activities of [counting, ordering, measuring (lengths, areas, volumes, times, masses), comparing in quantity/size, adding, taking away, distributing (which one does initially with particular things)] to numbers (as abstract things which can be manipulated) and arithmetic on numbers (like, one can now add 2 and 3, as opposed to only being able to put together a collection of 2 objects of some kind with a collection of 3 objects of another kind), which is a main contributor to e.g. opening up an arena of mathematical activities in elementary number theory, algebra, and geometry; one can get from one doing operations/calculations to/with numbers (like squaring the side length of a square to find its area) to the notion of a function (mapping numbers to numbers), which contributes to it becoming sensible to e.g. add functions, rescale functions, find fourier decompositions of periodic functions, find derivatives, integrate, find extrema, consider and try to solve functional equations; one can again turn aspects of these activities with functions into yet more objects of study, e.g. now getting vector spaces of functions, fourier series/transforms (now as things you might ask questions about, not just do), derivatives and integrals of functions, functionals, extremal points of functions, and differential equations, and looking at one’s activities with functions can e.g. let/make one state and prove the fundamental theorem of calculus and the extreme value theorem; etc.. (I think these examples aren’t entirely historically accurate — for example, geometric thinking and infinitesimals are neglected in the above story about how calculus was developed — but I think the actual stories would illustrate this point roughly equally well. Also, the actual process is of course not that discrete: there aren’t really clear steps of getting objects from activities and starting to perform new activities with these new objects.) (To avoid a potential misunderstanding: even though many things studied in mathematics are first invented/discovered/encountered “in” our (thinking-)activities, I do not mean to say that the mathematical study of these things is then well-seen as the study of some aspects of (thinking-)activities — it’s probably better-seen as a study of some concrete abstract things (even though it is often still very useful to continue to understand these abstract things in part by being able to perform and understand something like the activities in which they first showed up).) Philosophy also does a bunch of seeing things in thought and proceeding to talk about them; here’s an example stack: “you should do X”-activities -> the notion of an obligation -> developing systems of obligations, discussing when someone has an obligation -> the notion of an ethical system/theory -> comparing ethical theories, discussing how to handle uncertainty over ethical theories -> the notion of a moral parliament. Of course, science also involves a bunch of taking some stuff discovered in previous activities as subject matter — consider how there is a branch of physics concerned with lenses and mirrors and a branch concerned with electric circuits (with batteries, resistors, capacitors, inductors, etc.) or how it’s common for a science to be concerned with its methodology (e.g., econometrics isn’t just about running regressions on new datasets or whatever, but also centrally about developing better econometric methods) — but I guess that given that this step tends to take one from some very concrete stuff in the world to some somewhat more abstract stuff, there’s a tendency for these kinds of steps to exit science and land in math or philosophy (depending on whether the questions/objects are clearly specified or not) (for example, if an econometrician asks a clear methodological question that can be adequately studied without needing to make reference to some real-world context, then that question might be most appropriately studied by a [mathematical statistic]ian).)↩︎
I should maybe say more here, especially if I actually additionally want to communicate some direct intuition that the two things are equi-infinite.↩︎
this should maybe be split into more classes (on each side of the analogy between technological things and mathematical things)↩︎
one can make a further decision about whether to look “inside” the rationals as well here↩︎
incidentally, if you squint, Gödel’s completeness theorem says that anything which can be talked about coherently exists; it’d be cool to go from this to saying that in math, given any “coherent external structure”, there is an “internal structure”/thing which [gives rise to]/has that external structure; unfortunately, in general, there might only be such a thing in the same sense that there is a “proof” of the Gödel sentence \(G\) — (assuming PA is consistent) there is indeed an object \(x\) in some model of PA such that the sentence we thought meant that \(x\) is a proof of \(G\) evaluates to True, but unfortunately for our story (and fortunately for the coherence of mathematics), this \(x\) does not really correspond to a proof of \(G\).↩︎
One could try to form finer complexity classes (maybe like how there are different infinite cardinalities in set theory), i.e., make it possible to consider one infinite problem more infinite than another. I’d guess that the problems considered here would still remain equi-infinite even if one attempts some reasonable such stratification).↩︎
I think there are also many other important kinds of thinking-technologies — we’re just picking something to focus on here.↩︎
something like this was used for AlphaGeometry (Fig. 3)↩︎
For example, you might come up with the notion of a graph minor when trying to characterize planar graphs. The notion of a graph minor can “support” a characterization of planar graphs.↩︎
I find it somewhat askew to speak of structures “in the brain” here — would we say that first-order logic was a structure “in the brain” before it was made explicit (as opposed to a structure to some extent and messily present in our (mathematical) thought/[reasoning-practices])? But okay, we can probably indeed also take an interest in stuff that’s “more in the brain”.↩︎
I don’t actually have a reference here, but there are surely papers on plants responding to some signals in a sorta-kinda-bayesian way in some settings?↩︎
I’d also say the same if asked to guess at a higher structure “behind/in” doing more broadly.↩︎