todo post

Author: agarnung

_posts/XXXX-XX-XX-proximal-mapping.md

@@ -12,7 +12,7 @@ description: An introduction to proximal mappings and operators
 
 Los mapeos proximales son ampliamente utilizados en optimización convexa. Daremos un contexto teórico y los supeditaremos al procesamiento de imagen.
 
-## 2. Mapeo Proximal de la Norma \( \ell_2 \) 
+## Mapeo Proximal de la Norma \( \ell_2 \) 
 
 INSERTAR PDF
 

_posts/XXXX-XX-XX-quine.md

@@ -0,0 +1,108 @@
+---
+title: Quine 
+tags: [math, computer science, philosophy, divulgation]
+style: fill
+color: success
+description: Quine programs and self-referential computing
+---
+
+![image](./image.png)
+
+# Introduction
+
+[Quine (computing) - Wikipedia](https://en.wikipedia.org/wiki/Quine_(computing))
+
+## Implementation
+
+### Writing a Quine Program in C++
+```cpp
+```
+
+## Non-formal Mathematical Intuition
+
+INSERT PDF
+
+Let's put the previous document in more accessible terms. The basic idea is that we are constructing a **"manual of instructions for generating mathematical knowledge."**
+
+### 1. Axiom as the Fundamental Basis
+
+_"Every formal deductive system contains at least one undefined primitive element."_
+
+- Axioms are like the **atoms** of mathematical thought—truths accepted without proof.
+- Example in **Euclidean geometry**: _"A straight line can be drawn between any two points."_
+- Similarly, axioms function like the **basic rules of a board game**—they are not questioned but accepted as a foundation.
+
+### 2. Theorem as an Epistemic Morphism
+
+_"A theorem is a morphism in the category of formal proofs [...]"_
+
+- A **morphism** is a transformation between mathematical structures that preserves internal structure, like a recipe that converts ingredients into a cake while maintaining their essence.
+- **Domain (Axioms) → Codomain (Conclusions)**. Example:
+  - **Axioms**: Arithmetic rules
+  - **Theorem**: \( a + b = b + a \) (commutativity)
+  - **Morphism**: The logical process that derives this new truth from basic rules.
+- The **commutativity of the logical diagram** ensures that regardless of the logical path taken, the same conclusion is reached.
+
+### 3. Lemma of Conceptual Self-Support
+
+_"For every formal system \( \Sigma \), there exists an injective function between definitions and theorems..."_
+
+Breaking it down:
+- **Formal system (\( \Sigma \))**: A set of rules and symbols (like chess: pieces + movement rules).
+- **Injective function**: Each definition generates a unique theorem (like each key opens only one lock).
+- **\( \Sigma \cup \{ \Sigma \} \)**: A system that can describe itself (like a dictionary that includes the word "dictionary").
+
+Example:
+- **Definition**: _Even number_ = integer divisible by 2
+- **Generated theorem**: _The sum of two even numbers is even_
+- The function \( \iota \) ensures that this theorem directly **emerges** from the definition.
+
+### 4. Principle of Methodological Reflection
+
+_"There exists an elementary embedding between a metamodel and its consistent extension."_
+
+Discussion:
+- **Metamodel (𝔐)**: Like the blueprint of a building.
+- **Consistent extension (Con(𝔐))**: The actual constructed building, now including instructions for self-modification.
+- **Elementary embedding (j)**: Ensures that everything true in the blueprint remains true in the expanded structure.
+
+Example: Imagine a **video game that can modify itself while running**, without breaking its original rules.
+
+### 5. Fundamental Theorem of Self-Referential Exposition
+
+\[ \Sigma \vdash \text{Con}(E_{\Sigma}) \leftrightarrow \text{Con}(\Sigma) \]
+
+- **\( \Sigma \)**: Your mathematical theory (e.g., number theory)
+- **\( \text{Con}(\Sigma) \)**: "\( \Sigma \) does not allow contradictions"
+- **\( E_{\Sigma} \)**: A user manual for \( \Sigma \) that includes instructions on how to use \( \Sigma \)
+- **Equivalence**: If your theory is consistent, then its **"user manual"** is also consistent, and vice versa, avoiding the direct self-reference that Gödel blocks in his **Incompleteness Theorem**.
+
+The act of explaining a mathematical theory does not introduce errors **if the original theory was solid.**
+
+### 6. Corollary of Structural Integrity
+
+\[ E_{\text{Comp}}(E) = E_{\text{Prof}}(E) \otimes E_{\text{Clar}}(E) \]
+
+- **\( E_{\text{Comp}} \)**: Overall quality of exposition
+- **\( E_{\text{Prof}} \)**: Technical depth
+- **\( E_{\text{Clar}} \)**: Expository clarity
+- **\( \otimes \)**: Harmonic combination (not a simple sum)
+
+Analogous to **Einstein’s equation** \( E = mc^2 \), where the quality of exposition arises from the interaction between:
+- **Mass (depth)**
+- **Light (clarity)**
+
+### 7. The Quinean System (Final Observation)
+
+**Key Concept:**
+
+- **Coherent self-reference**: The document uses the concepts it defines, creating a **loop of validity.**
+
+- **Literary example**: A novel where the protagonist writes the novel you are reading.
+
+### Logical Guarantee
+Consistency emerges from the relationship between:
+
+- **Syntax**: How symbols are written
+- **Semantics**: What those symbols mean
+