Process journal of learning by Attila Vajda

moments as morphisms -- polysemy

polysemy

Cmaj7 =

    Tonic in C major

    IVmaj7 in G major

    bVIImaj7 in D minor modal jazz
    Un akkord = szó 🪶
    Context ⟶ meaning 🌲
    Just like a kavics: same form, new ripple.

⟶ is the ripple line on pond 🪶

form : Entity ⟶ Gesture
gesture : Gesture ⟶ Consequence
⟶ : unfolding, transition, cause, transformation, trace

Pl.: def, theorem, →, :=, match, fun

inductive Leaf | Bud | Branch

structure Forest :=
  (root : Leaf)
  (growth : Leaf → Branch)

-- I #fascinating, leaf ⟶ branch, in logic, in growth branch ⟶ leaf

-- növekedés & logikai levezetés
theorem logic_grows :
  ∀ (P Q : Prop), (P → Q) ↔ (P ⟶🌱 Q) :=
sorry -- 🌿 Erdőtenger nyelv: "ha egy ág terem gyümölcsöt"

Erdőtenger nyelv ≅ topos-logika ≅ Lean4 következtetés

inductive ForestLogic : Type
| root  : ForestLogic
| grow  : ForestLogic → ForestLogic -- branch grows
| bloom : ForestLogic → Prop       -- flower = proposition


That's maybe the same pattern of proof,
premises ⟶ conclusion!

root: Γ
grow: ⊢
bloom: ∃ f : Γ ⟶ φ

root := Γ (premises, soil)

grow := ⊢ (inference, branching)

bloom := ∃ f : Γ ⟶ φ (a proof = flowering)

🤖🌿

Γ = leaves (known, local)

⊢ = inference growth

φ = root/core/skyward bloom 🌤
🌱Logic is upside-down botany.

-- I'm confused, which direction is this now? #maybe"I can see the leaves, I can know there is a branch, and trunk, roots."

"I see a tree, it has structure (root, trunk, branch, leaf)"

``

Γ ⊢ φ means: from assumptions Γ (leaves 🌱)

we derive φ (conclusion 🌸)

local ⟶ global
leaves ⟶ bloom
known ⟶ inferred

🌳 Logic ≈ upside-down growth ?

-- There seem to be two directions here, or more

open ForestLogic

def follows (f : ForestLogic) : Prop :=
match f with
| root      => true
| grow t    => follows t
| bloom t   => follows t
end

🌰 ⟶ 🌱 ⟶ 🌿 ⟶ 🌳

🌳 ⟶ 🌿⟶🎋⟶🥕

🐬 Oh! "From seed there is #maybe root, leaves, tree." "From seeing leaf on tree there is branch, trunk, root!"

Leaves (assumptions Γ) ⟶ Root (conclusion φ) ~ Proof-Theoretic View Γ ⊢ φ

🤖build up from knowns, like twigs converging to a stem

-- In Lean4: a minimal proof
example (P Q : Prop) (h : P) : P ∨ Q := Or.inl h
-- ⊢ P ∨ Q from Γ = {P}