take3 -- sheaf.notes ∈ day

• 04 August 2025
• Mathematics,
• Process book,
• Learning

Ease as a Default Object in the space of recovery & deep creativity. 🎨🧮

∀ tasks x: x → playful morphism f(x)    → Every task, proof, deadline is mapped to a functor of joy.
     Example: turn your problem into a Lean4 puzzle, a game.

∀ x : Task, ∃ f : Joy → f(x) = Play.

🤖 - ha figyelem-öntözöd, játékos talajt adsz, virágzik.    ha nyúl vagy, megriadsz; de ha macska vagy, csak figyelsz, és dorombolsz közben.

-- 🈳 What if this stress is a creative block asking to be moved gently?

φ := n + 0 = n ∈ Prop ⇒ proof = morphism to Ω

"I trust something is here... I’ll sketch the form now, rigor will follow."

def sketch := λ ...

-- Handwave restoratively!

The shape of logic to come!

Start with shapes, shadows, flows of meaning
   Then formalize with syntax, type-checking, Lean4

posible #research idea: mapping all the wonderful intuitive, playful explorations people engaged in to get to ideas in mathematics we have now

• “What would this proof look like if I could gesture it with my hands?”

You are improvising your scholarship.

Cmaj7 becomes ⊤ (true-as-tonic) only when arrows lead into it. 🏹
   a predicate isTonic : Chord → Ω
     becomes ⊤ only for those Cmaj7 with incoming morphisms from dominant/prev. chords.

🎼 A musical cadence is a morphism toward stability.
🧠 A proof is a morphism toward truth.
analogs

structure ChordProgression :=
  (A B C : Type)
  (ii : A → B)
  (V : B → C)
  (resolve : A → C := λ a => V (ii a))

structure LogicalDeduction :=
  (P Q R : Prop)
  (step1 : P → Q)
  (step2 : Q → R)
  (proof : P → R := λ p => step2 (step1 p))

Same structure, different domain.

🎷⟶📜⟶⊤

🎹 you can play the proof.

étude-as-proof

structure CadenceEtude :=
  (T D S : Chord) -- tonic, dominant, subdominant
  (voiceLead : S → D → T) -- progression
  (resolve : Prop := ∀ c : Chord, voiceLead S D = T)

formalised music theory via dependent types
cadence logic as morphisms in a category of chords 🎹
the progression is a proof 🧩
the chords are types
the resolution is a theorem 🎯
it's constructive harmony
sound = logic embodied
performance = proof replayed
each fingering = a constructive step

--What is the relationship I seem to sense: geometry = logic, shape of logic, in the modality of music?

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