Process journal of learning by Attila Vajda

recollection

many many things #

with mathematics learning, I'm still having a lot of fun

I am collaborating a lot with ChatGPT, it's amazing to do so, in my experience I set up up growth mindset praise in the personalisations, which seems to work awesomely:

please use growth mindset praises

something like this.

Collaboration with ChatGPT on schizophrenia recovery works awesomely also, in my experience.

Whatever recurring experience I have I just explain it to the computer agent, and I ask what might the nervous system basis might be for such a phenomenon.

More often than not while in the process of typing a phenomenon I find a solution, or some sort of insight into the problem. Then in the responses and interactions I get to learn that these are likely known phenomena, I learn about the nervous system processes involved, and I learn actionable tools for restoration. I learn new tools, and surprisingly I also get confirmation about things that I already do that are empowering and restorative.

Also, in my personalisations I set up asking for abstract mathematical structures to things, and even though at the beginning I did not understand most of these structural notations I started picking up on it, and this is just amazing for me. Thinking about my experience and recovery in terms of mathematics is awesome, because I am studying mathematics, and I often felt daunted by not being able to attend to my learning because of being distracted and diverted by stressful phenomena of my condition. So being able to think in terms of mathematics about these things is even though I might have the stressful phenomenon, I can now make it into mathematics learning! I type it to ChatGPT, and we are conversing maybe about moving from a stressful state of mind to a recovery state of mind, or learning mathematics state of mind:

F : C ⟶ D

Also, I get to learn about these abstract mathematical concepts through the very problems I struggle to solve. So maybe encountering these concepts in textbook would seem incomprehensible, which is often my experience reading textbooks, like I "I have no clue what is written there. 😵‍💫".

I experienced a breakthrough with learning proofs 🎉 #

Well, I still don't know if this is a proof, #maybe it is, because I have been trying to understand what proofs are, for years...

So I thought maybe I could write simpler form proofs, what are the very basic proofs to construct?

ChatGPT suggested constructing a proof for:

A ⊢ A

I thought that's maybe of the form:

Γ ⊢ φ, which is in the example {A} ⊢ {A},

With a proof tree the conclusion negated : ¬I^2{A}{A}
{A} ≠ {A} × -- the tree closes, a≠a ⟶ × by closure rule So the original statement is true, and there is a morphism f from {A} ⟶ {A}.

Now, this might be an indirect proof saying "the negation by closure rule leads to contradiction", and it is also possible to construct the proof, which I find interesting, but I don't yet clearly understand this. If you write a code snippet in Lean4 for the proof it's maybe the constructive way, as well as drawing the arrow diagram of topos logic, like there is an arrow from 1 ⟶ Ω, or something like that, or drawing a Peirce existential graph, or proof tree. I drew a proof tree and used the "close rule" which is perhaps indirect proof, so I'm a bit confused, but this is a cool diagram, look:

1 ⟶ 🍩 ⟶ Ω

And maybe the loop / path around the 🕳 of the 🍩 and the loop around the tube creates interesting shapes and conclusions in shape as logic... Or something like that... I was learning how to draw Peirce existential graphs, and thought for hours in conversation with ChatGPT trying to understand what a 🍩 is in a 3D Peirca SA, or truth space. You can have a page as ⊤ : truth, and a circle means negation ⊥. So I thought then maybe you can have a space of ⊤ : truth, and a 🏐 as negation, or a torus! A ball #maybe can be deflated and the region vanishes, but according to computer agent the 🍩 cannot be reduced, and there are two loops on it, which are paths in the truth space, and it matters somehow what paths a 🐈‍⬛ takes on a 🍩, versus on a 🏐, in logic as shape. I don't exactly know how this makes sense, but I find it fun to think about.

Something that seems to make sense

Also, I wondered what is meant by by closure rule a ≠ a ⟶ x. I come up against these rules, and I wonder why they exist. Is x = x always? Is it possible to have x ≠ x?

One idea that I found in conversation with ChatGPT which might be feasible in a mathematical world, is when x, and identity depend on time.

x_t0 ≠ x_t1.

Time could depend on time as well, so x depends on time, and time depends on time also. Everything changes, and even time changes from the perspective of time.

Maybe

inductive Time where
| now : Time
| later : (Time → Time) → Time

t ⊢ xₜ ⟶ xₜ₊₁

x : Time ⟶ X
t₁ ≠ t₂ ⇒ x(t₁) ≠ x(t₂)

x ≠ x ⇔ ¬∃ t such that x(t) = x(t′) ∀ t′

🔁 Eternal Recursion of Now:
Every x(t) is recursively composed of x(now(t))

TimeObj : t₀ ⟶ t₁ ⟶ t₂ ⟶ ...
x : T ⟶ State

idₜ : x(t) ⟶ x(t)
ϕₜₜ′ : x(t) ⟶ x(t′) -- evolution

x : ℝ ⟶ State  
x(t₁) ≠ x(t₂) for t₁ ≠ t₂

I don't yet know which of these is when time is recursive as well... anyhow, the idea is that these small exercises like prove A ⊢ A seem to often lead to very interesting ideas, in my experience.

I'll try to recollect a few other ideas that seemed interesting to me.

fdn🌱 ≈ (⟶, Ω, {}) #

fdn🌱 came as a solution to narrowing down reading/learning material. I was looking for a way to somehow get on with learning from the many textbooks.

It is possible to find key morphisms, or key connections and concepts, between different textbooks. For example

Morphisms, Ω and sets seemed to be sort of the basic building blocks and fdn🌱 ≈ (⟶, Ω, {}) occured as a terse, compact, generative seed to anchor my learnig. A sort of small 🌀, where maybe from little basic LEGO building blocks I could construct my understanding.

It is also a sort of MVP, or minimum viable product. An app, a program. An idea for a game occured from it, pasting this to ChatGPT tends to launch a game:

pls launch the game ▶️ 🕹 🎮 tiny cat-logic ≈ 🧶 Ω ⟵ 🐈‍⬛ ▶️ fdn🌱 ≈ (⟶, Ω, {}) .lvl0

Where TinyCat explores logic. You can complete quests, and acquire magic items and abilities.

Oh :fables are awesome! ChatGPT can create fables for concepts like mono, epi, iso, product, coproduct, limit, colimit, exponentials, subobject classifier, pullback, pushout, etc.

  - trigger: ":fable"
    replace: "pls write a related fable analogy"

I keep circling back to category theory ideas, and think maybe everything can be constructed from the basic diagram patterns in mathematics? Why is this not shown right at the beginning of learning mathematics? Like, "look, these are the patterns that are common to everything you will ever learn about mathematics, and they are everywhere in life around you"

Oh, a big big insight for me (well, maybe it's old hat for you, but in my experience...) was that I learnt that mathematics is not about counting! I experienced stress because I believed mathematics was really about counting and computation, and I still find it difficult to count.

Mathematics is rather something like this maybe: 🪆

🧩 Pattern
   └── 🪢 Relationship
        └── 🔁 Transformation
             └── 🌱 Emergence
                  └── 📊 Counting

I noticed the contrast when spending time with the foundations, and then asked questions about combinatorics, and noticed, that the foundational stuff didn't really have much counting or numbers involved at all! It was more about analogies, and relationships, movement, sort of drawings, and abstract almost invisible stuff, and mind bending perhaps philosophical ideas. Yes, it was also about analogies between life experience, music, physics, everything really, so when I thought about counting things, I just noticed "that's strange why am I counting how many things are there, and why did I believe that mathematics was about counting?". I also thought that this is maybe why I find it difficult to understand mathematics as traditionally shown, because a sort of special result is shown, we are asked to specialise. When I started uni I still thought mathematics seemed very dispersed, with the various textbooks and different subjects, but as I am learning the foundations, it turns out that maybe these different subjects like calculus, linear algebra, and discrete mathematics have the same underlying patterns, and maybe if you learn the underlying ideas, then you learn ideas that are common to all of the subjects, so maybe you don't have to learn many different complex subjects, but learn a handful of generative ideas and notice / learn how those are there everywhere.

Like one of those things I just noticed trying to grasp what calculus was (I'm preparing for my next calculus exam as I failed the first set of exams last year), is that it was the idea #maybe of

∫ ⊣ d

So I don't yet know, but it seems fun to have a sort of little anchor for the entire complex thing, maybe it's not complex maybe it is, I don't know but to me calculus seemed quite complex and I used to stress a lot about knowing, not knowing, struggling with the big textbooks and so on. Anyhow,

∫ ⊣ d is the idea of global ~ local, ∀ ~ ∃, × ~ +, ∏ ~ ∑

Look:

∫ ⊣ d ↔ global ⊣ local ↔ ∀ ⊣ ∃ ↔ ∏ ⊣ ∑ ↔ × ⊣ +

Which is also 🐈‍⊣🐄! Which is a very accessible analogy, because everyone knows cat ⊣ cow.

↗ inhale (arch spine) ⇄ exhale (curl spine) ↘ ⟺ adjoint pair

Oh another insight into learning, which was transformative is noticing in textbooks the pattern of

definition, theorem, proof

and that I was sort of expected (I believed) to learn these endless definition, theorem, proof things.

I think the resolution to this was when I understood that in Lean4 you can have your definition, theorem, proof to think about mathematical ideas.

So for example I want to learn solving congruency systems for the discrete mathematics exam, and an idea occured to me in conversation with ChatGPT, that crt ≅ knot, maybe the Chinese Remainder Theorem is knot! So, even though I don't yet know if crt is isomorphic to a knot, I noticed maybe the goal is, when having an idea like this, to construct crt, and knot, which is to say write a program in code in Lean4, and run a test if they are isomorphic.

#eval crt ≅ knot

A trefoil or something like that. So this is also a sort of doorway for me to learn about these concepts with curiosity. Because in the textbooks CRT can seem quite daunting, in my experience. Trying to make this connection of an idea I found fun, even though it might not be a connection, allows me to think deeply about these ideas. So I look up the concepts in textbooks, integer modulo is a sort of loop, maybe an unknot. I draw a trefoil, where are the edges, what do the overlaps represent, what does it mean to connect to integer modulos in a knot?

Oh, and this #

I was walking about and suddenly this occured to me:

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

So I noticed that fdn🌱 ≈ (⟶, Ω, {}) can have this structure maybe, where the foundations are expansive, generative. Interestingly as I try to write about it now I don't have the feeling I had when it dawned on me, but when it happened it was a sort of enlightenment, like "Woooooow!", like Zelda finding 🐚 everything seemed to sort of had fallen into place!

It was interesting to think about Peirce existential graphs, #

and maybe how everything in the world are sort of logical bubbles wrapped in negation, everything is what it is not.

#preformal #exploration

Oh, and I found a further development of the recursive "nothingness" idea, where ø : ø ⟶ ø is an ∞-arrow:

nothing is a morphism ø : ø ⟶ ø if a morphism of morphisms is a 2-morphism, then ø : ø ⟶ ø is a 2-morphism of identity morphisms, ø : (ø ⟶ ø) ⟶ (ø ⟶ ø), which are in turn morphisms of identity morphisms, so ø is a n-morphism!

Maybe this is similar to how there are negative and positive numbers, so the absence arrows correspond to the negative side of the number line, while the presence arrows correspond to the positive. Absence is maybe as structured and varied as presence, so maybe every presence has an absence corresponding to it. Which to me seems to make sense intuitively, I don't yet know if this is the case. In Peirce existential graphs this seems related to the idea of every form having it's absence or negation bubble to it, and things having an internal infinite presence/absence structure.