2 infinitude spreads

  1. An endeavor being infinite often causes other related endeavors to be infinite. In particular:
    1. If \(E\) is an infinite endeavor, then “how should one do \(E\)?” is also infinite.1 For example: math is infinite, so “how should one do math?” is infinite; ethics is infinite, so “how should one do ethics?” is infinite.2
    2. If \(E\) is an infinite endeavor and there is a “faithful reduction” of \(E\) to another endeavor \(F\), then \(F\) is also infinite. (In particular, if an infinite endeavor \(E\) is “faithfully” a subset of another endeavor \(F\), then \(F\) is also infinite.)3 For example, math being infinite implies that stuff in general is infinite; “how should one do math?” being infinite implies that “how should one think?” is infinite.
    3. If an endeavor constitutes a decently big part of an infinite endeavor, then it is infinite.4 5 For example, to the extent that language is and will remain to be highly load-bearing in thinking, [figuring out how thinking should work] being infinite implies that [figuring out how language should work] is also infinite.
  2. Thinking being infinite can help make some sense of many other philosophical problems/endeavors being infinite.6
    1. Specifically:
      1. If “solving” a philosophical problem would entail understanding some aspect of thinking which has a claim to indefinitely constituting a decently big part of thinking, then
      2. If “solving” a philosophical problem would entail making a decision on [how one’s thinking operates in some major aspect], that philosophical problem is not going to be remotely “solved”, because one will probably want to majorly change how one’s thinking works in that major aspect later.
    2. Here are some problems whose infinitude could be explained by the infinitude of thinking in this way (but whose infinitude could also be explained in other ways):
      1. “how does learning work?”
      2. “how does language work?”
      3. “how should one assign probabilities?”
      4. “what are concepts? which criteria determine whether concepts are good?”
      5. “how does science work?”
      6. “how should one do mathematics?”
      7. “what is the character of value(ing)?”
    3. You might feel indifferent about some of these problems; you could well have a philosophical system in which some of these are not load-bearing (as I do). My claim is that those problems which are genuinely load-bearing in your philosophical system are probably infinite, and their infinitude can be significantly explained by the infinitude of “how should one think?”.
    4. We could imagine a version of these problems which could be solved — for example, one might construct a GOFAI system which has a (kinda-)language+meaning which is plausibly load-bearing for it but which will not be reworked. My claim is that for a mind which will indefinitely be thinking better, if it has “language” and “meaning”, to the extent these are load-bearingly sticking around indefinitely, the mind will also “want to” majorly rework them indefinitely.
    5. Again, all this is very much not to say that one cannot make progress on these philosophical problems — even though any progress will be infinitesimal for these infinite endeavors, one can make substantive progress on them (just like one can make substantive progress in math), and I think humanity is in fact continuously making substantive progress on them. So, for many philosophical problems/endeavors \(P\), I can get behind “we’re pretty much no closer to solving \(P\) now than we were 2000 years ago” — meaning that only an infinitesimal fraction has been solved, just like earlier, or that only a finite amount has been figured out, with a real infinity still remaining, just like earlier — while very much disagreeing with the possible followup “that is, we haven’t made any progress on \(P\) since 2000 years ago” — I think probably more than half of the philosophical progress up until now happened since 1700, and maybe even just since 1875.7
  3. Saying these problems are infinite because of the infinitude of “how should one think?” can make it seem like we are viewing ourselves very much “from the outside” when tackling these problems and when tackling “how should one think?”, but I also mean to include tackling these problems more “from the inside”.
    1. What do I mean by tackling these problems “from the outside” vs “from the inside”? Like, on the extreme of tackling a problem “from the outside”, you could maybe imagine examining (yourself as) another thinking-system out there, having its language and beliefs and various thought-structures somehow made intelligible to you, with you trying to figure out what eg its concept of “existence” should be reworked to. Tackling “the same problem” from the inside could look like having various intuitions about existence and trying to find a way to specify what it means for something to exist in terms of other vocabulary such that these intuitions are met (this can feel like taking “existence” to already be some thing, just without you having a clear sense of what it is). And a sorta intermediate example: asking yourself why you want a notion of “existence”, trying to answer that by examining cases where your actions (or thinking more broadly) would depend on that notion, and seeking to first make sense of how to operate in those cases in some way that routes through existence less. Of course, in practice, we always have/do an amalgam of the inside and outside thing.8
    2. I’m saying that the infinitude of [such investigations which are more from the inside of a conceptual schema] could also be partly explained by the infinitude of “how should thinking work?”.9
  4. Wanting to rework one’s system of thought indefinitely is also a reason for keeping constituent structures provisional.

onward to Note 3!

  1. The weaker claim that I could make here is that this is only true typically — i.e., that “if \(P\) is infinite, then”how should one do \(P\)?” is usually also infinite. I think the weaker claim would be fine to support the rest of the discussion, but I’ve decided to go with the stronger claim for now as it still seems plausible. To adapt an ancient biology olympiad adage: it might be that all universally quantified statements are false, but the ones that are nearly true are worth their weight in gold. Anyway, I admit that the claim being true ends up depending on how one measures progress on a problem (as discussed briefly also in the last item), which I haven’t properly specified. For these notes, I would like to keep getting away with saying we’re measuring progress in some intuitive way which is like the way mathematicians measure progress when saying math is infinite.↩︎

  2. I think the metaethical problem of providing sth like axioms for ethics is also infinite, despite the fact that it could be reasonable to consider the analogous problem for math finite given that we can develop (almost) all our math inside [first-order logic]+ZFC. One relevant difference between this metamathematical problem and this metaethical problem is that any system which supports sufficiently rich structure can be a fine solution to the metamathematical problem because we can then probably make \(\approx\)all our mathematical objects sit inside that system, whereas it is far from being the case that anything goes to this extent for the metaethical problem — it being fine to use the hypothetical ethical system to guide action feels like a much stricter requirement on it, and in fact probably entangles the choice with ethics sufficiently to make this problem infinite, also. Additionally, it being imo plausibly infinite to make some decision on what the character of value is also pushes this metaethical problem toward infinitude. What’s more, such a formal system would not only need to “contain” our values, but also our understanding (I will talk more about this in a later note), but our understanding is probably not something on which any fixing decision should ever be made because we should remain open to thinking in new ways, so any hypothetical such system would be going out of date after it is created (at least in its “understanding-component”, but I also doubt a principled cleavage can be made (I’ll discuss various reasons in upcoming notes)). Also, our understanding and values at any time \(t\) are also probably much too big for us to see properly at that time \(t\), as well as not in a format to fit in any canonically-shaped such system. That said, there is a “small problem of providing an ethical system” which just asks for any kind of “system” which is sorta fine to give control to according to one’s values (relative to some default), which is solvable because e.g. humanity-tomorrow will be such a system for humanity.↩︎

  3. By \(E\) reducing to \(F\), I roughly mean the usual thing from computer science: that there is a cheap/easy way to get a solution to \(E\) from a solution to \(F\); this implies that if you could solve \(F\), you could also solve \(E\). If we conceive of each endeavor as consisting of collecting/solving some kinds of pieces, then we could also require a piece-wise map here: that for each piece of \(E\), one can cheaply specify a piece of \(F\) such that having solved/collected that piece of \(F\) would make it cheap for one to solve/collect that piece of \(E\) as well. I’m adding the further condition that the reduction be “faithful” to rule out cases where a full solution to \(F\) would give you a solution to \(E\), but one can nevertheless get a positive fraction of the way in \(F\) without doing anything remotely as challenging as \(E\) — for example, take \(F\) to be the disjoint union of playing a game of tic-tac-toe and math, with equal total importance assigned to each, and take \(E\) to be math. I’m not sure what I should mean by a reduction being “faithful” (a central criterion on its meaning is just to make the statement about transferring infinitude true), but here are a few alternatives: (1) If we can think of \(E\) and \(F\) in terms of collecting pieces with importance-measures in \(\mathbb{R}^{\geq 0}\), then it would be sufficient to require that the reduction map doesn’t distort the measures outrageously. (2) We could say that any \([\geq\frac13]\)-solution of \(F\) would cheaply yield a \([\geq c]\)-solution (with \(c>0\)) of \(E\), or that any way to remotely-well-solve \(F\) would cheaply give rise to a way to remotely-well-solve \(E\), but this is plausibly too strong a requirement. (3) We could say that if there were a \([\geq\frac13]\)-solution of \(F\), there would be another solution of \(F\) which isn’t “infinitely many times more challenging (to find)” which would cheaply yield a \([\geq c]\)-solution (with \(c>0\)) of \(E\). In other words, we’d say that there is a reasonably economical way to go about \([\geq \frac13]\)-solving \(F\) which would involve \([\geq c]\)-solving \(E\) as well. Actually, for most infinitude-transfers one might want to handle with this rule, I think it might also be fine to do without a “precise” statement, just appealing to how \(F\) is intuitively at least as challenging as \(E\).↩︎

  4. Any non-infinitesimal fraction should definitely count as a “decently big part”; the reason I didn’t just say “positive fraction” to begin with is that I’d maybe like this principle to also helpfully explain why some sufficiently big infinitesimal fractions of infinite problems are infinite.↩︎

  5. a footnote analogous to the first footnote about the first item on this sublist↩︎

  6. Is the infinitude of “how should one think?” the “main reason” why philosophy is infinite? Is it the main reason for most particular infinite philosophical problems being infinite? I would guess that it is not — that there are also other important reasons; in particular, if a philosophical problem is infinite, I would expect there to at least also be some reason for its infinitude which is “more internal to it”. In fact, I suspect that clarifying the [reasons why]/[ways in which] endeavors can end up infinite is itself an infinite endeavor :).↩︎

  7. In addition to stuff done by people canonically considered philosophers (who have also made a great deal of progress), I’d certainly include as central examples of philosophical progress many of the initial intellectual steps which led to the creation of the following fields: probability and statistics, thermodynamics and statistical mechanics, formal mathematical logic, computability and computer science, game theory, cybernetics, information theory, computational complexity theory, AI, machine learning, and probably many others; often, this involved setting up an arena in which one could find clear/mathematical counterparts to some family of previously vague questions. (I’d also consider many later intellectual steps in these fields to be philosophical progress.)↩︎

  8. Doing some amount of the inside thing is hard to avoid because we can’t access ourselves except from the inside, and even if we had some other access channel, we’d struggle to operate well on an aspect of ourselves except by playing around with it on the inside a bunch — think also about eg how to help someone out of a confusion (about a theorem, say), it can help to try to think in that confused way yourself, and eg how playing a game (or at least seeing it played) is centrally good for improving it. Some amount of the outside thing is also usually unavoidable — we are reflective by default and that’s good — think eg about how it is actually good/necessary to also look at a game from the outside to improve it, not to just keep playing it (though a human will of course be doing a great deal of looking at the game even when just playing it), and about why it’s good that we are able to talk about propositions and not solely about more thingy things (or, well, about why it’s good we can turn propositions into ordinary thingy things)8.↩︎

  9. Really, I’m somewhat unhappy with the language I have here — “how should thinking work?” sounds too much like we’re only taking the external position. I would like to have something which makes it clearer that we have here is like a game at once played and improved.↩︎