3 math, thinking, and technology are equi-infinite

  1. If one of math, understanding intelligence, and inventing useful technologies is infinite, then so are the other two. (An argument for this is given in items 2–6 of this list.)
    1. So, if you think any one of the three is infinite, then you should also think that the other two are. In particular, if you’re on board with my earlier assertion that math is infinite, then you should also agree that thinking and technology are infinite. Or you could prima facie buy into technology being infinite, and get to thinking that thinking and math are infinite from there.
  2. If math is infinite, then “how should one do math?” is infinite.
    1. I could just appeal to note 2 item 1.1 to justify this, but it makes sense to also justify this independently (especially given that I haven’t justified 2.1.1 well). (Also, if we went with the definition of an infinite endeavor in 1.4.3(.1), then we’d be done by definition here, but that’s not too exciting, either.)
    2. I guess I’d first like to try to get you to recognize that if you accepted my earlier assertion that math is infinite, then you might have already sorta accepted that doing math is infinite, at least in the following sense:
      1. If we think of math-the-more-object-level-endeavor as being ultimately about printing proofs of propositions, then we should plausibly already think of say, mathematical ideas (e.g., the idea of a probabilistic construction) or mathematical objects/constructions (e.g., a vector space) or frames/arenas organizing mathematical fields (e.g., the scheme-theoretic organization of algebraic geometry) as meta-level things — they are like tools one uses to print proofs better.
      2. This isn’t to say that math is in fact ultimately well-thought-of as not also being about defining objects and coming up with new ideas and so on. But I want to get across the sense that these live in significant part also on the meta-level — that they are not just things in the object-domain, but also components of us doing math. Getting further in math centrally involves gaining these tools, from others or by making them yourself.1
    3. Quite generally, given that understanding a domain better blends together with thinking better in the domain to a significant extent, if you think there’s an infinitely rich variety of things to understand in mathematics, this gives you some reason to think that there is also an infinitely rich variety of structure to employ in doing mathematics, so it gives you some reason to think that figuring out how one should do math is an infinite endeavor.
  3. If “how should one do math?” is infinite, then “how should one think?” is also infinite.
    1. We could just say this is a special case of 2.1.2 (i.e., item 1.2 from Note 2), but it also makes sense to justify this separately/again. I think it’d be super bizarre for mathematical thinking and thinking in general not to be equi-infinite; it’d be particularly bizarre for getting better at thinking to be infinitely “easier” than getting better at mathematical thinking. They are just way too similar (in the relevant aspect(s)). It’s not like math researchers use thinking in some profoundly different class when doing math (compared to when doing other thinking). Things in general are way too much like math way too much of the time. Doing things in general involves doing math way too centrally.
  4. If “how should one think?” is infinite, then tech is also infinite.
    1. The question “how should one think?” is/[centrally involves] the question “which thinking-[structures/technologies] should one make/use?”, so this latter question is then also infinite (maybe by 2.1.3). And “thinking-technologies” are (“faithfully”) a subset of all technologies, so tech is also infinite by 2.1.2.2
  5. Finally, if tech is infinite, then math is infinite.
    1. The vibes are very similar, in the relevant way.
    2. In particular, technological objects are a lot like mathematical objects. From this, one might think that tech is like a fragment of math, in which case one might think that math could only be more infinite than tech, appealing to 2.1.2.
    3. But also, our mathematical ideas “show up load-bearingly” all over the place in our technologies, so one could try to see math as a fragment of tech and to infer its infinitude from the infinitude of tech using 2.1.3.
    4. Actually, our math just is one kind of tech we’re building (for our thinking and also for other doings) — this gives another way to see math as a subendeavor of tech for us, and one can then again try to infer its infinitude using 2.1.3.
    5. Additionally, math shows up load-bearingly in developing tech, so one could infer the infinitude of developing tech from the infinitude of tech using 2.1.1 and then infer the infinitude of math from that using 2.1.3.
    6. Let us say more about math and tech having similar vibes (in the relevant way). One could tell many stories to communicate that the vibes are similar; here are the premises of a few:
      1. Mathematical objects are often invented/discovered to do things, to [participate in]/support/[make possible] our (mathematical) activities, and to help us with problems/projects — just like (other) technological objects.
      2. One can think of a technology in terms of its construction/composition/design/[internal structure], like an engineer might, or one can think of a technology in terms of [its function(s)]/[its purpose(s)]/[what one can do with it]/[how to use it]/[what it is like as it operates]/[its properties]/[how it relates to other things]/[the surrounding external structure]3, like one usually does when thinking about doing stuff with the technology. This is also true of mathematical devices — for example, one can think of the real numbers in terms of their construction (e.g. as Cauchy sequences of rationals4 or Dedekind cuts of rationals), or one can instead think of the real numbers as a complete ordered field, or as being involved in geometry and analysis in various ways, etc.; one can think of the homology groups of a topological space in terms of their construction (e.g. via the simplicial route or the singular route), or one can instead think of homology groups as things of which the various theorems in Hatcher involving homologies are true (indeed, one can pin down the homology of a space uniquely as the thing satisfying some axioms — see Section 2.3 and Theorem 4.59 in Hatcher5); one can think of a product of topological spaces as having tuples as points and having its topology generated by cylinder sets, or as a canonical topological space equipped with continuous maps to the original spaces.
      3. One can also just be using a technology without thinking about it much (for example, you can be using your laptop to write notes without thinking about your laptop much), and one can also be using a mathematical thing in one’s thinking without thinking about it much (for example, you can be doing stuff with real numbers without explicitly thinking about how their construction or even their properties). (However, a difference: there is structure supporting/[involved in] your use of your laptop which lives outside your head, whereas the structure involved in your use of a vector space lives \(\approx\)entirely in your head.)
  6. Note that the implications about infinitude in items 2-5 on this list form a loop, so we’ve provided an argument that these endeavors are equi-infinite. More precisely, we’ve given an argument that if any one of these problems/endeavors is infinite, then any other is also infinite; this means that they are either all finite or all infinite.6
  7. You might say: “Okay, I agree that there’s a genuine infinity of mathematical objects (and so on), a genuine infinity of technologies, and a genuine infinity of thinking-technologies in particular. But couldn’t there be a level above all these infinite messes where there’s some simple thing?”. My short answer is: no, e.g. our handling of mathematical objects is technological/organic throughout. A longer answer:
    1. For example, let’s look at making concepts.7 Making concepts looks e.g. like having a varied system for morphological derivation and inflection, giv-ing parti-cul-ar ab(i)-(bi)li(s)-ti-es to pro-duce con-cep-t-s. Here are some more example high-level ways in which new concepts can be found:
      1. become familiar with many toy cases; push yourself to see them as clearly as possible; if any objects show up, try to see if they can also be used in other contexts;
      2. when you have a toy setting worked out, ask what “auxiliary constructions”/“nontrivial ideas” were involved8;
      3. generalize existing things; unify existing things; articulate similarities; find what’s responsible for similarities;
      4. make existing looser notions more precise; make distinctions;
      5. get concepts for a novel context via analogy to some familiar context;
      6. look for a thing which would “work” a certain way; look for a thing which would play a certain role9;
      7. more generally, have some constraints on a hypothetical thing in mind and look for a concrete thing which would satisfy those constraints;
      8. in particular, for various particular relations or functions, look for what stands in that relation with some thing or what that function takes some thing to;
      9. in particular, look for the (logical) cause of some property or event;
      10. look for ways you can relate something to other things; look for things you can do with it; look for ways to transform it;
      11. enumerate all things of some kind and see if any are useful.
    2. Making concepts can look a lot like making (more explicit) technologies, in which surely very much of our thinking is importantly involved.
    3. Also, I’d sorta object to thinking there is a level “above” this messy world of concepts. There are surely structures present in us-doing-math beyond easily visible mathematical concepts, including structures of different kinds, but instead of thinking of these structures as handling and organizing concepts from above, I think it’s better to think of the whole thinking-shebang as like an organic thing, so this sounds e.g. like thinking that [blood circulation]/[the cardiovascular system] is a structure “above” the organs, or like saying a tree is a structure organizing koalas. Additionally, the structure surrounding a concept is in significant part [given by]/[made of] other concepts.
    4. You might say: “hmm, you’re talking about all these things we can easily see, but couldn’t there be a nice hidden structure which handles things? like, a structure in the brain?”.
      1. Well, I certainly have been talking about those things we can see better, and there are surely many structures in our thinking-activities10 that we can’t see that clearly at present. It seems unlikely that there’d be these central hidden things of a fundamentally different character than the various technological things we can see, though.
    5. You might say: “hmm, maybe thinking is this organic-technological mess now, but couldn’t it become well-organized? couldn’t there be some sort of formula for thinking which remains to be discovered/reified?”. Or maybe you might say: “hmm, maybe thinking is an organic-technological mess, but couldn’t it really be a shadow of a very nice thing?”
      1. There are surely nice things shadowed in thinking. For example, (almost all) clear human mathematical statements and clear mathematical proofs (especially from after like 1950 or whatever) have formal counterparts in ZFC — there’s a very real “near-isomorphism” of an important thing in human mathematical (thinking-)activities with a nice formal thing. For another example, there are ways to see bayesianism (usually combined with other interesting “ideas”) in various things from (frequentist) statistical methods to plant behavior11. It can often be helpful to understand something about the actual thing by evoking the nice thing; it is in some contexts appropriate to think almost entirely about a nice thing to figure out something about the actual thing. But I think there are very many nice things shadowed in thinking, with developing thinking continuing to shadow more cool things on all levels indefinitely.
      2. This seems close to asking whether the (technological) world would eventually just “have” one kind of thing. And it seems unlikely that it would!
      3. There will probably continue to be a very rich variety of (thinking-)structures in use.
      4. If I had to guess at some single higher “structure” centrally “behind/in” thinking, I’d say “one is creative; one finds/invents and finds uses for (thinking-)structures”.12 But I don’t think this is some sort of definite/simple thing — I think this will again be rich, involving many “ideas”.
      5. I’ve tried to say more in response to these questions in the next few notes.

onward to Note 4!


  1. 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).)↩︎

  2. I should maybe say more here, especially if I actually additionally want to communicate some direct intuition that the two things are equi-infinite.↩︎

  3. this should maybe be split into more classes (on each side of the analogy between technological things and mathematical things)↩︎

  4. one can make a further decision about whether to look “inside” the rationals as well here↩︎

  5. 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 \(G\).↩︎

  6. 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).↩︎

  7. I think there are also many other important kinds of thinking-technologies — we’re just picking something to focus on here.↩︎

  8. something like this was used for AlphaGeometry (Fig. 3)↩︎

  9. 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.↩︎

  10. 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”.↩︎

  11. 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?↩︎

  12. I’d also say the same if asked to guess at a higher structure “behind/in” doing more broadly.↩︎