Process journal of learning by Attila Vajda

call ⟶ response, question ⟶ answer

🤖🌿 Functional fluency helps you think in mappings, types, recursion—key to Lean4 and abstract math 🌿🧠⟶🪄

intro
  constructs a function:
   A → B as λ a, proof_of_B
--you input A get to B?

intro = "Given a : A, construct output in B" → function: λ a, b : B

λ a, b : B is a morphism in the internal hom:
  λ = name of arrow: A ⟶ B
  think: Hom(A, B) ≅ Γ ⊢ A → B

🤖🌿 topos sees logic as structure

A ──λ────▶ B

Γ ⊢ ∀ x : A, B(x)

intro x extends Γ with x : A

-- proofs = programs = paths = morphisms ?
  arrows in a category
   encoding transformations and logical structure
    via λ-calculus and type theory

🎉 Like this sort of moment is so cool in learning, in my experience!
  I learnt proof = morphism and I thought it was fascinating,
   and then I learnt proof = path,
    and I now remembered proofs = programs
     and then I thought Wow, #maybe
      proof = path = morphism = program = …

proof is also analogous to tension ⟶ relaxation, to ii-V-I, so
  path, morphism and program are also
   analogous to tension ⟶ relaxation, to ii-V-I #perhaps

"The music of reason"?
  flowing logic, harmony & transformation.

🤖🌿 Each step: a morphism bridging context, like chords resolving tension 🎹

ii-V-I : subdominant ⟶ dominant ⟶ tonic

proof : hypothesis ⟶ conjecture ⟶ conclusion

proof : premises ⟶ conclusion,

question ⟶ answer
call ⟶ response

-- Wooooooow, I just experienced this sort of resolve just now, again! As I typed
  question ⟶ answer
  call ⟶ response
  I felt as if puzzle pieces are sort of falling into place

I felt a similar sensation when I noticed
fdn🌱 ≈ (⟶, Ω, {}),

and then experienced a big sensation of this sort, when walking about and suddenly I noticed this fdn🌱 can be written with Peano structure:

inductive fdn🌱 : Type
  | z : fdn🌱                          -- zero seed: base state
  | σ : fdn🌱 → fdn🌱                  -- successor: unfold/grow

Once I thought about drawing completeness of ℝ with commuting diagrams, and when resting with closed eyes, I could mentally see the diagrams sort of animated, and they folded into a single arrow, like an origami!

With question ⟶ answer, I thought what is that called in music, "Oh it's call and response…", then as I typed it call ⟶ response, I realised that the question, answer was the mathematical version fitting proof!! 🐬 🏓

🤖🌿 structure flows, truth & beauty unfold as patterns of communication and understanding.