by providing a picture of waldo in the cut-out, you can prove you know where he is without providing the location. a zero knowledge proof.
Where’s Waldo as presented isn’t even a proof of knowledge
1. THEOREM: Let a semantic frame be defined as Ω = (Σ, R), where
Σ is a finite symbol set and R is a finite set of inference rules.
Let Ω′ = (Σ′, R′) be a candidate successor frame.
Define a frame jump as: Frame Jump Condition: Ω′ extends Ω if Σ′\Σ ≠ ∅ or R′\R ≠ ∅
Let P be a deterministic Turing machine (TM) operating entirely within Ω.
Then: Lemma 1 (Symbol Containment): For any output L(P) ⊆ Σ, P cannot emit any σ ∉ Σ.
(Whereas Σ
= the set of all finite symbol strings in the frame; derivable outputs are formed from Σ under the inference rules R.)Proof Sketch: P’s tape alphabet is fixed to Σ and symbols derived from Σ. By induction, no computation step can introduce a symbol not already in Σ. ∎
2. APPLICATION: Newton → Special Relativity
Let Σᴺ = { t, x, y, z, v, F, m, +, · } (Newtonian Frame) Let Σᴿ = Σᴺ ∪ { c, γ, η(·,·) } (SR Frame)
Let φ = “The speed of light is invariant in all inertial frames.” Let Tᴿ be the theory of special relativity. Let Pᴺ be a TM constrained to Σᴺ.
By Lemma 1, Pᴺ cannot emit any σ ∉ Σᴺ.
But φ ∈ Tᴿ requires σ ∈ Σᴿ \ Σᴺ
→ Therefore Pᴺ ⊬ φ → Tᴿ ⊈ L(Pᴺ)
Thus:
Special Relativity cannot be derived from Newtonian physics within its original formal frame.
3. EMPIRICAL CONFLICT Let: Axiom N₁: Galilean transformation (x′ = x − vt, t′ = t) Axiom N₂: Ether model for light speed Data D: Michelson–Morley ⇒ c = const
In Ωᴺ, combining N₁ and N₂ with D leads to contradiction. Resolving D requires introducing {c, γ, η(·,·)}, i.e., Σᴿ \ Σᴺ But by Lemma 1: impossible within Pᴺ. -> Frame must be exited to resolve data.
4. FRAME JUMP OBSERVATION
Einstein introduced Σᴿ — a new frame with new symbols and transformation rules. He did so without derivation from within Ωᴺ. That constitutes a frame jump.
5. FINALLY
A: Einstein created Tᴿ with Σᴿ, where Σᴿ \ Σᴺ ≠ ∅
B: Einstein was human
C: Therefore, humans can initiate frame jumps (i.e., generate formal systems containing symbols/rules not computable within the original system).
Algorithmic systems (defined by fixed Σ and R) cannot perform frame jumps. But human cognition demonstrably can.
QED.
BUT: Can Humans COMPUTE those functions? (As you asked)
-> Answer: a) No - because frame-jumping is not a computation.
It’s a generative act that lies outside the scope of computational derivation. Any attempt to perform frame-jumping by computation would either a) enter a Goedelian paradox (truth unprovable in frame),b) trigger the halting problem , or c) collapse into semantic overload , where symbols become unstable, and inference breaks down.
In each case, the cognitive system fails not from error, but from structural constraint. AND: The same constraint exists for human rationality.
This is really sloppy work, I'd encourage you to look deeper into how (eg) HOL models "theories" (roughly corresponding to your idea of "frame") and how they can evolve. There is a HOL-in-HOL autoformalization. This provides a sound basis for considering models of science.
Noncomputability is available in the form of Hilbert's choice, or you can add axioms yourself to capture what notion you think is incomputable.
Basically I don't accept that humans _do_ in fact do a frame jump as loosely gestured at, and I think a more careful modeling of what the hell you mean by that will dissolve the confusion.
Of course I accept that humans are subject to the Goedelian curse, and we are often incoherent, and we're never quite surely when we can stop collecting evidence or updating models based on observation. We are computational.
Just to be clear: it's not "associated" -- the drug itself probably smells, and you just didn't notice it. It's very, very common with drugs containing amines.
Metformin, for instance, has a distinctly fishy odor (but it also has a couple more amines):
https://en.wikipedia.org/wiki/Metformin#/media/File:Metformi...
the fishy smell is not characteristic of pure amphetamine, but leftover methylamines from synthesis.
vyvanse adds a lysine but none of these amines are free. it’s odorless as well, but any lysine esthers leftover will stank.
it was shit product
As a total tangent: it would be interesting to have an LLM-based modality, like a browser extension, where a user could highlight academic concepts in a pdf and drill down. Academic writing, by convention and necessity, is terse and references prior literature, sometimes opaquely. So getting up to speed in the literature takes significant effort.
and it is so wonderful for it:)
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did it cure my depression? nah