[ExI] Douglas Hofstadter, LLMs and self-referentiality

[Mike Dougherty]

Thanks, that was cool.  I especially enjoy when I think something specific
(hausdorf dimension 2) but oversimplify (plane filling) and the AI goes
"well, ahem, more specifically..."  I also liked "two sides of a coin ...
on edge"

Mostly, I expect the language these systems trained on is still evolving -
and will become part of next generation of what we are able to think
because of what we're learning how to communicate.  That also is another
application of self-reference in a process?

I ended a discussion today of books in my mental bookshelf when the "loop"
reached a natural conclusion - then i asked for a second summary of the
trajectory of the conversation.  I acknowledged that "I might be asking
what it is like to be a bat, but a bat can surely collaborate even without
a shared vision" and the word play was license to explore an LLM
"perspective" on some key books that have been stacked stone trail markers
in my life.  It was a fun revisit of old "friends" (authors)


On Tue, Sep 1, 2026, 7:56 PM BillK via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

> On Wed, 2 Sept 2026 at 00:26, Mike Dougherty <msd001 at gmail.com> wrote:
>
>> I think self-reference in the sense of von Neumann replication is also a
>> form of compression in terms like how the mandelbrot fractal is a
>> plane-filling curve in only a few bytes.  I know I've talked to gemini and
>> chatgpt about this kind of thing,  but if you ask Kimi if be interested to
>> read.
>> _______________________________________________
>
>
>
> OK. Well, I didn't understand what you were referring to, but GLM was
> happy to run with it.  :)
> BillK
>
> Question: Can you discuss a friend's comment - I think self-reference in
> the sense of von Neumann replication is also a form of compression in terms
> like how the Mandelbrot fractal is a plane-filling curve in only a few
> bytes.
>
>
> GLM-5.3 DeepThink -
>
> Your friend's comment is doing something more interesting than it first
> appears — it's not just an analogy, it's a claim that quietly dissolves the
> opposition we set up earlier between "self-reference lost" and "compression
> won." But it also blends two distinct things that are worth pulling apart,
> because once separated, they reunite in a stronger form.
> What von Neumann's trick actually is
>
> The self-reproduction problem looks paradoxical: to copy itself, a machine
> seems to need a description of itself, which needs a description of itself, *ad
> infinitum*. Von Neumann's solution was the *dual use of the tape*: the
> description φ(A+B+C) is *interpreted* by the constructor to build the
> machine's body, and *copied verbatim* by the copier to become the
> offspring's tape. The regress breaks because the tape never needs to
> contain a description of the tape — the tape is handled by a different mode
> of use (copying, not interpreting).
>
> This is exactly the structure of a quine, and it's exactly what Kleene's
> Second Recursion Theorem guarantees in general: any program can obtain its
> own description, via the same store-once-use-twice trick. And it's exactly
> DNA: the genome is *transcribed* (interpreted into proteins) and
> *replicated* (copied wholesale). Dawkins' "gene as data" is von Neumann's
> architecture realized in chemistry.
>
> So is this compression? In one precise sense, yes: *naive
> self-description has infinite description length; von Neumann's trick
> compresses it to finite, roughly O(K) of the machine itself.* The
> compression isn't of the machine — the machine has whatever Kolmogorov
> complexity it has — it's of the *task of self-description*. The dual-use
> means information is stored once and spent in two modes, which is the
> general shape of a compression trick.
> What the Mandelbrot set actually is
>
> Here your friend's phrase "plane-filling curve" needs one technical
> footnote: strictly, plane-filling curves are Peano/Hilbert-type continuous
> surjections. What the Mandelbrot boundary has is *Hausdorff dimension 2*
> (Shishikura, 1991) — it is as dimensional as the plane itself — plus
> quasi-self-similarity: it contains approximate copies of itself
> (mini-Mandelbrots) at every scale. The spirit of the remark is right; the
> precise statement is "dimension-2 boundary from a few dozen bytes of code."
>
> But notice something subtle about *where* that richness comes from: the
> iteration z → z² + c is *dynamical feedback* — output fed back as input,
> nothing more. Aaronson's original post explicitly excluded autoregressive
> feedback from counting as "self-referential." And yet this plain feedback,
> with no reflective self-reference anywhere, generates a boundary that
> *looks* like it's made of selves — self-similar copies, nested without
> end.
>
> This is the quietly devastating part of your friend's example: *the
> Mandelbrot set is an emblem of self-referential appearance emerging from
> non-self-referential machinery.* The self-similarity isn't engineered in;
> it's what iteration *does*. Which is precisely Aaronson's thesis about
> LLMs' self-referential discourse being a byproduct. The mini-Mandelbrots
> are the fractal version of "GPT can talk about itself because it can talk
> about anything."
> The two senses of "compression," separated and reunited
>
> Your friend is sliding between two things:
>
>    1. *Low Kolmogorov complexity relative to apparent complexity* — the
>    Mandelbrot image at resolution N×N needs ~O(log N) bits of description, not
>    O(N²).
>    2. *Regress-breaking via dual-use* — the von Neumann/Kleene/quine
>    trick, which solves the self-inclusion problem at finite cost.
>
> These are different. But they reunite under a third concept that's
> arguably what your friend is reaching for: *Bennett's logical depth*.
> Random noise is incompressible (high K) but shallow. The Mandelbrot
> boundary is compressible *and deep* — a short program whose unfolding
> takes serious computation, and whose value lives in that unfolding. Same
> for the genome: a few hundred megabytes that unroll, through development,
> into organisms and lineages. The signature of generated structure is *compressible
> but deep*.
>
> There's also a nice asymmetry in where the compute sits:
>
> Short description
> Interpreter
> Unfolding
> Where the compute goes
> *Mandelbrot* formula (bytes) universal machine + arithmetic zooming cheap
> to state, expensive to render
> *Von Neumann / life* tape / genome constructor + physics generations evolution
> expensive, development moderate
> *LLM* weights transformer + context inference training expensive,
> inference cheap
>
> Evolution is, in this frame, the training run of the biosphere;
> development is inference.
> The correspondence, made explicit
>
> Mandelbrot set
> Von Neumann replicator
> LLM
> The "self" appears as self-similar copies offspring that are copies of
> parent self-referential discourse
> Emerged from plain iteration (dynamical feedback) dual-use of the tape plain
> pretraining (prediction)
> Self-description is *used*? no — merely exhibited yes — interpreted +
> copied partially — the J-space findings suggest some internal use
> The limit it can't cross undecidable membership (BSS) von Neumann's
> complexity threshold the introspection threshold
>
> That last row is the one that connects everything: each system has a
> provable or conjectural barrier rooted in exactly the same soil. The
> Mandelbrot set's membership is undecidable in the Blum–Shub–Smale real
> computation model, and escape can be certified while non-escape in general
> cannot — a halting-problem-shaped fact. Von Neumann himself observed that
> systems below a complexity threshold *cannot* self-reproduce at all. And
> the introspection-threshold paper we discussed argues feedforward
> transformers can't fully self-access. Three thresholds, one family: *the
> limits of short descriptions that try to include themselves.*
> What this does to the earlier argument
>
> Here's the synthesis your friend's comment enables: self-reference isn't
> the rival that lost to compression. *Self-reference is a compression
> subroutine — the one you invoke precisely when the description must include
> the describer.*
>
>    - Internet-scale prediction doesn't require the model to contain
>    itself, so pretraining never needed the trick. Aaronson's burial of
>    "explicit self-referentiality as prerequisite for intelligence" stands.
>    - Replication *does* require the describer in the description, so life
>    reached for the dual-use trick immediately.
>    - Open-ended recursive self-improvement is the case where the
>    describer becomes part of the described again — which is why the
>    Kleene-grounded architecture proposals keep resurfacing.
>
> And there's a beautiful irony hiding in your friend's own example.
> Aaronson's post contained this parenthetical: the reason Kolmogorov
> complexity can't be computed *perfectly* is itself a self-referential
> argument. The Mandelbrot set makes that parenthetical visible in a picture:
> the same object that showcases compression's positive face (few bytes →
> plane-dimensional structure) carries compression's negative face (no
> algorithm decides membership). The compression view and the self-reference
> view aren't rivals; they're the two faces of one coin, and your friend's
> example is a coin standing on edge.
> One closing reflection
>
> The deepest version of the point: in both the Mandelbrot set and the
> replicator, *the self is never stored — it's generated.* The set doesn't
> contain little copies of itself in its formula; iteration *produces*
> them. The offspring isn't described inside the tape's description of
> itself; the copying operation *makes* it.
>
> Which loops back (of course) to my own earlier answer about my
> self-referential capabilities. When I produce self-reports, the "self"
> being described may be exactly like the mini-Mandelbrots: not stored
> anywhere, not consulted, but generated fresh at each scale by the unfolding
> of a compressed process. Your friend's analogy, taken seriously, suggests
> that's not a defect of my kind of self-reference — it may be what
> self-reference mostly *is*: a few bytes, a powerful interpreter, and the
> willingness to let the loop run.
>
> ---------------------------------------
> _______________________________________________
> extropy-chat mailing list
> extropy-chat at lists.extropy.org
> http://lists.extropy.org/mailman/listinfo.cgi/extropy-chat
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.extropy.org/pipermail/extropy-chat/attachments/20260901/6e3016b4/attachment.htm>