Yudhister Kumar
This argues that utilitarianism is selfish egoism, but not the contrary? My reading of your position is that someone who had a utility function not dependent on the wellbeing of any other beings would be a selfish egoist, but it’s difficult for me to understand how that could be utilitarian.
idiolects?
French fluency is neither necessary nor sufficient for understanding EGA.
There’s a certain sense in which understanding a particular French “dialect” (the collection of words + localized grammar + shared mental context required to make sense of EGA, the one which forms the basis for modern French algebraic geometry (?)) is a sufficient condition for understanding EGA.
There’s also a sense in which understanding this French algebro-geometric dialect is an almost necessary condition for understanding EGA past a certain point (happy to consider disputations, and perhaps the understanding one receives from the necessity condition is less directed at the concepts which the literature built off of but rather the peculiarities of Grothendieck et. al.’s mental states & historical context).
Packaging “shared mental context” with a “dialect” and subsequently claiming that understanding the “dialect” is necessary and sufficient for understanding the embedded concepts is begging the question.
It seems like there is this restricted language associated with a set of concepts, the concepts themselves can are understood in the context of the restricted language, the concepts are mostly divorced from the embedded grammar of the parent language, and we don’t have a very good way of drawing a boundary around this “restricted language.”
In a general sense, this kind of “conceptual binding” is not rigid. Strong Sapir-Whorf is incorrect, the Ghananian can learn English, I can just read Hartshorne or solely Anglophonic literature to learn algebraic geometry.
However, canonical boundaries make sense even when the the boundaries are leaky. A species is not completely closed under reproduction, however it makes sense to think of species as effectually reproductively closed. A cell wall separates a cell from its environment, even if osmosis or active transport allows for various molecules to be transported in and out.
One might expect this binding to be “stronger” when the inferential distance between the typical concepts of some reference class of language-speaker and the concepts discussed in the “dialect” to be larger.
A general description of a language used by a group of communicators is the tuple (alphabet, shared conception of grammatical rules, shared semantic conception of language atoms & combinator outputs).
Outside of purely formal settings, the shared conceptions of grammar & semantics will be leaky. How much can be purely recovered from shared words?
However, there are natural attractors in this space. Ex. traditional dialects, modern languages. Shared conception diffs between language-speakers are significantly smaller than shared conception diffs between two different language speakers (this is by default unresolvable unless there’s some shared conception of translation, at which point they’re sort of speaking the same conceptual language?)
When talking about algebraic geometry, it feels like an English geometer and a French geometer are speaking more similar languages than a French geometer and a French cafe owner.
I want to say: “an idiolect is a natural attractor in the space of languages for a group of communicators discussing a certain set of concepts, the idioms of the idiolect are identified with the concepts discussed, and the idiolect is quasi-closed under idiomatic composition.”
Identifying shared languages as emergent coordination structures between a group of communicators feels satisfying.
However, returning to the case of algebraic geometry, it feels like I can “grok” the definitions of the structures described without understanding the embedded French grammar in EGA. Maybe the correct decomposition of a shared language is (shared idiomatic conception) + (translation interface), and we should just care about the “pre-idiolect.”
This is just a world model? Describable without reference to other communicators? Loses some aspect of “coordination”?
Maybe the pre-idiolect is s.t. n communicators can communicate idioms & their compositions with minimal description of a translation interface.
The idiom <-> concept correspondence feels correct. Like, on some level, one of the primary purposes of a grammatical structure is to take the concepts which are primarily bound to words & make sense of their composition, and lexicogenesis is a large part of language-making. But it feels like restricting to wordly atoms is too constraining and there are structural atoms that carry semantic meaning, and idiom can encompass these.
How do you reify concept-space enough to chunk it into non-overlapping parts?
I am trying to point at a superstructure and say “here is a superstructure.” I am trying to identify the superstructure by a closure criterion, and I am trying to understand what the closure criterion is. Something language-like should be identifiable this way? And the appropriate notion of closure will then let us chunk correctly?
Maybe superstructures are not generally identifiable via closure?
The load-bearing constraint for considering species as superorganisms is a closure property. They’re not particularly well-describable by Dennett’s intentional stance.
I want to say “idiolect:species :: communicator:member-organism :: idiom:gene.”
I don’t want to identify lexemes as the atoms of a language-like-structure. Chomsky et. al.’s new mathematical merge formalism is cool but construed, and I have not seen a clean way to differentiate meaningful lexeme composition from non-meaningful lexeme composition.
“Shared understanding” feels better? The point of a language is a mechanism by which communicators communicate, and it so happens that languages happen to be characterizable by some general formal propeties.
Really appreciated this!
Cosma Shalizi just posted a similar list: http://bactra.org/notebooks/math.html
yeah this is straightforwardly wrong, thanks. the first part should be read like “this is a way you can construct a physical realization of an automata corresponding to a type-3 grammar, this is in principle possible for all sorts of them”
will get back to you with something more rigorous
(very naive take) I would suspect this is medium-easily automatable by making detailed enough specs of existing hardware systems & bugs in them, or whatever (maybe synthetically generate weak systems with semi-obvious bugs and train on transcripts which allows generalization to harder ones). it also seems like the sort of thing that is particularly susceptible to AI >> human; the difficulty here is generating the appropriate data & the languages for doing so already exist ?
but only the dialogues?
actually, it probably needs a re-ordering. place the really terse stuff in an appendix, put the dialogues in the beginning, etc.
I’m less interested in what existing groups of things we call “species” and more interested in what the platonic ideal of a species is & how we can use it as an intuition pump. This is also why I restrict “species” in the blogpost to “macrofauna species”, which have less horizontal gene transfer & asexual reproduction.
I haven’t looked much at the extended phenotype literature, although that is changing as we speak. Thanks for pointing me in that direction!
The thing I wanted to communicate was less “existing groups of things we call species are perfect examples of how super-organisms should work” and more “the definition of an ideal species captures something quite salient about what it means for a super-organism to be distinct from other super-organisms and its environment.” In practice, yes, looking at structure does seem to be better.
Payor’s Lemma holds in provability logic, distributivity is invoked when moving from step 1) to step 2) and this can be accomplished by considering all instances of distributivity to be true by axiom & using modus ponens. This section should probably be rewritten with the standard presentation of K to avoid confusion.
W.r.t. to this presentation of probabilistic logic, let’s see what the analogous generator would be:
Axioms:
all tautologies of Christiano’s logic
all instances of (weak distributivity) --- which hold for the reasons in the post
Rules of inference:
Necessitation
Modus Ponens
Then, again, step 1 to 2 of the proof of the probabilistic payor’s lemma is shown by considering the axiom of weak distributivity and using modus ponens.
(actually, these are pretty rough thoughts. Unsure what the mapping is to the probabilistic version, and if the axiom schema holds in the same way)
No particular reason (this is the setup used by Demski in his original probabilistic Payor post).
I agree this is nonstandard though! To consider necessitation as a rule of inference & not mentioning modus ponens. Part of the justification is that probabilistic weak distributivity () seems to be much closer to a ‘rule of inference’ than an axiom for me (or, at least, given the probabilistic logic setup we’re using it’s already a tautology?).
On reflection, this presentation makes more sense to me (or at least gives me a better sense of what’s going on / what’s different between logic and logic). I am pretty sure they’re interchangeable however.
Yep! Thanks!
Yep! Fixed.
We know that the self-referential probabilistic logic proposed in Christiano 2012 is consistent. So, if we can get probabilistic Payor in this logic, then as we are already operating within a consistent system this should be a legitimate result.
Will respond more in depth later!
Language mix-up. Meant improper integrals.
Now that I’m thinking about it, my memory’s fuzzy on how you’d actually calculate them rigorously w/infinitesimals. Will get back to you with an example.
Have updated the definition of the derivative to specify the differences between over the hyperreals and over the reals.
I think the natural way to extend your to the hyperreals is for it to take values in an infinitesimal neighborhood surrounding rationals to 0 and all other values to 1. Using this, the derivative is in fact undefined, as
I agree. This is what I was going for in that paragraph. If you define derivatives & integrals with infinitesimals, then you can actually do things like treating dy/dx as a fraction without partaking in the half-in half-out dance that calc 1 teachers currently have to do.
I don’t think the pedagogical benefit of nonstandard analysis is to replace Analysis I courses, but rather to give a rigorous backing to doing algebra with infinitesimals (“an infinitely small thing plus a real number is the same real number, an infinitely small thing times a real number is zero”). *Improper integrals would make a lot more sense this way, IMO.
Here the magic lies in depending on the axiom of choice to get a non-principal ultrafilter. And I believe I see a crack in the above definition of the derivative. f is a function on the non-standard reals, but its derivative is defined to only take standard values, so it will be constant in the infinitesimal range around any standard real. If , then its derivative should surely be everywhere. The above definition only gives you that for standard values of .
Yep, the definition is wrong. If then let denote the natural extension of this function to the hyperreals (considering behaves like this should work in most cases). Then, I think the derivative should be
W.r.t. what the derivative of should be, I imagine you can describe it similarly in terms of , which by the transfer principle should exist (which applies because of Łoś′s theorem, which I don’t claim to fully understand).
For the derivative then is:
I’m familiar with \setminus being used to denote set complements, so \not\in seemed more appropriate to me ( is not an element of ). I interpret as “the elements of not in ,” which is the empty set in this case? (also the elements of are sets of naturals while the elements of are naturals, so it’s unclear to me how much this makes sense)
Noting this was posted on April 1, as it won’t be immediately apparent to posterity.