Sets with dependent elements: A formalization of Castoriadis' notion of magma

This paper presents a formalization of Cornelius Castoriadis' concept of "magmas" within a strengthened ZFA set theory by modeling them as nonempty open subsets of a pre-ordered set of atoms, thereby constructing a hierarchical "magmatic universe" where elements depend on one another in a manner distinct from standard Cantorian sets.

The Gist

Here is an explanation of the paper "Sets with dependent elements: A formalization of Castoriadis' notion of magma," translated into everyday language with creative analogies.

The Big Idea: The "Sticky" Universe

Imagine you are used to thinking about the world like a box of LEGO bricks. In a standard LEGO box, every brick is independent. You can pick up a red 2x4 brick, put it in a pile, take it out, or throw it away, and the other bricks don't care. They are separate, distinct, and free. This is how traditional mathematics (Cantor's Set Theory) views the world: everything is a collection of independent, distinct items.

The philosopher Cornelius Castoriadis argued that this view is wrong for many things in real life. He pointed to things like language, memories, or culture.

• The Problem: You can't really separate the word "love" from the word "heart" or the concept of "romance." If you try to isolate the word "love" in your mind, it drags all those other connected ideas along with it. They are "sticky." They depend on each other.
• The Concept: Castoriadis called these sticky, interconnected collections "Magmas." A magma isn't just a pile of separate things; it's a web where pulling one thread pulls the whole net.

This paper asks: Can we do math on these "sticky" things? Can we build a formal system (a set of rules) to handle collections where elements need each other to exist?

The Solution: The "Dependency Web"

The author, Athanassios Tzouvaras, says "Yes," but we have to change the rules of the game. Here is how he does it, using simple analogies:

1. The Atoms (The Raw Material)

Imagine a vast, infinite ocean of raw ideas or "atoms." Let's call this ocean A.
In normal math, these atoms are just floating around, unrelated. But in this paper, the author gives them a special property: Dependence.

He draws a map of the ocean where some points "point to" others.

• Analogy: Imagine a game of "Telephone." If you hear a whisper (Atom B), you immediately think of the person who told you (Atom A).
• The Rule: If Atom A depends on Atom B, you can't have A without B. They are linked. The author calls this a "pre-ordering" (a fancy way of saying a map of who depends on whom).

2. The First Layer: The "Open Nets" (M1)

Now, we start building our Magmas. A Magma is a collection of these atoms. But because they are sticky, you can't just grab a random handful.

• The Rule of the Net: If you grab a specific atom, you must also grab everything it depends on.
• Analogy: Imagine fishing with a net. If you catch a big fish (Atom A), the net is designed so that if you pull A up, all the smaller fish (Atoms B, C, D) that A is tangled with must come up too. You can't catch just the big fish alone.
• In math terms, these collections are called "Open Sets." They are "open" because they are always connected to the things below them. You can never have a "minimal" magma (a tiny, isolated piece) because every piece is tied to something else.

3. The Tower of Magmas (The Hierarchy)

This is where it gets cool. The author builds a tower, level by level.

• Level 1: The sticky collections of raw atoms (The Nets).
• Level 2: Now, imagine the Nets themselves are the new "atoms." Can we make a collection of Nets? Yes! But the same rule applies. If you pick a Net, you must pick all the Nets that are "inside" or "dependent" on it.
• Level 3: Now we make collections of Level 2 collections.
• The Result: You get an infinite tower (M1, M2, M3...) where every level is made of "sticky" collections of the level below it.

The Magic Trick: The author discovered a mathematical trick (called "shifting") that allows him to treat these complex, sticky levels using the standard rules of "subset" (being inside something else). It's like having a universal translator that lets you speak "Standard Math" even when you are talking about "Sticky Magmas."

What Does This Prove?

The author checks if his new "Magmatic Universe" follows the rules Castoriadis proposed.

1. No Isolation: You can never find a single, tiny, independent piece. (True: In this system, everything is infinite and connected).
2. No Cutting: You can't cut a Magma into two separate, independent pieces. (True: If you try to split a "Net," the pieces are still tangled).
3. The Mix: The universe contains both normal "independent" sets (like a box of LEGO) and "sticky" Magmas. (True: The system allows for both).

Why Should You Care?

This paper is a bridge between Philosophy and Math.

• Philosophy: It gives a rigorous mathematical structure to the idea that human thoughts, languages, and societies aren't just piles of separate facts. They are living, breathing webs where one idea triggers another.
• Math: It challenges the idea that the only way to do math is with independent, isolated objects. It suggests we can build a math for "thick," interconnected realities.

The Catch (The Author's Honest Conclusion)

The author admits something important at the end: Castoriadis himself might hate this.
Castoriadis believed that standard math (Set Theory) was part of the problem because it forces us to see the world as separate, independent things. By using Set Theory to define Magmas, the author is using the "enemy's weapon" to fight the enemy.

However, the author argues: "I had no other tools." He used the old, rigid logic to build a model of a flexible, sticky world. He hopes this is just a first step, and maybe one day we will invent a whole new kind of logic that doesn't need to force "stickiness" into "independent boxes."

Summary Metaphor

Think of the universe as a spiderweb.

• Standard Math sees the web as a collection of individual, separate threads.
• Castoriadis' Magma sees the web as a single, vibrating entity where touching one thread vibrates the whole thing.
• This Paper builds a mathematical model of that vibration, showing how you can count and organize the web without breaking the vibration, proving that "sticky" things can be studied scientifically.

Technical Summary

Here is a detailed technical summary of the paper "Sets with dependent elements: A formalization of Castoriadis' notion of magma" by Athanassios Tzouvaras.

1. Problem Statement

The paper addresses the philosophical and mathematical challenge of formalizing magmas, a concept introduced by the social philosopher Cornelius Castoriadis.

• The Core Issue: Standard Cantorian set theory assumes that elements of a set are distinct, definite, and ontologically independent. One can remove an element without affecting the existence of others.
• Castoriadis' Critique: Castoriadis argued that many real-world collections (e.g., the "totality of meanings" in a language, or a person's "totality of memories") do not fit this model. In these collections, elements are interdependent; an element $a$ cannot exist in the collection without the simultaneous presence of elements it depends on. Consequently, the concept of a singleton set $\{a\}$ is often meaningless in these contexts.
• The Goal: To provide a rigorous mathematical formalization of these "dependent collections" (magmas) within a set-theoretic framework, capturing the property that elements cannot be isolated from their dependencies.

2. Methodology

The author employs a modification of ZFA (Zermelo-Fraenkel set theory with Atoms/Urelements) to construct a "magmatic universe."

A. Theoretical Framework

• Base Theory: The work is conducted in ZFAD, a strengthening of ZFA.
• Atoms ($A$): The foundation consists of an infinite set of atoms $A$ (urelements), representing the "raw material" of the magma (e.g., basic meanings or memories).
• Dependence Relation ($\preccurlyeq$): A primitive binary relation is introduced on $A$.
  • Properties: It is a pre-ordering (reflexive and transitive).
  • Non-Minimality Condition (Axiom D3): Crucially, the relation must satisfy $(\forall a \in A)(\exists b \in A)(b \prec a)$. This ensures there are no "minimal" elements; every atom depends on at least one other atom. This prevents the existence of isolated singletons.

B. Topological Construction

• Lower Topology: The author utilizes the lower topology induced by $\preccurlyeq$ on $A$.
  • A subset $x \subseteq A$ is "open" (a magma at level 1) if it is downward closed: if $b \in x$ and $a \preccurlyeq b$, then $a \in x$.
  • The class of level-1 magmas is defined as $M_1(A) = LO(A, \preccurlyeq)$, the set of all non-empty open subsets of $A$.
• Exponential Shifting: To build higher levels of the hierarchy, the pre-ordering is "shifted" to the power set.
  • For $x, y \subseteq A$, define $x \preccurlyeq^+ y$ iff $(\forall a \in x)(\exists b \in y)(a \preccurlyeq b)$.
  • Key Theorem: When restricted to the class of magmas $LO(A, \preccurlyeq)$, the shifted relation $\preccurlyeq^+$ coincides exactly with the subset relation $\subseteq$.
  • This allows the recursive definition of the hierarchy using only the subset relation for subsequent levels.

C. The Magmatic Hierarchy ($M_\alpha$)

The universe of magmas is constructed inductively over ordinals $\alpha \ge 1$:

1. Base: $M_1(A) = LO(A, \preccurlyeq)$.
2. Successor: $M_{\alpha+1}(A) = LO(M_\alpha(A), \subseteq)$. (The set of non-empty subsets of $M_\alpha$ that are open with respect to inclusion).
3. Limit: $M_\alpha(A) = \bigcup_{\beta < \alpha} M_\beta(A)$.
4. Total Universe: $M(A) = \bigcup_{\alpha \ge 1} M_\alpha(A)$.

3. Key Contributions and Results

A. Formalization of Dependence

The paper successfully translates the philosophical notion of "dependence" into a topological property (downward closure). The condition that there are no minimal elements in the pre-order ensures that:

• All magmas are infinite.
• No magma is a singleton.
• No magma is $\subseteq$-minimal (every magma contains a proper sub-magma).

B. Verification of Castoriadis' Principles

The author reformulates Castoriadis' five principles (M1–M5) to avoid contradictions found in the original text (specifically M1 and M4). The formalization validates three modified principles within the structure $\langle M(A), \in \rangle$:

1. M2 (Sub-magma existence):* For every magma $x$, there exists a distinct magma $y \subset x$. (True because no set in the hierarchy is minimal).
2. M3 (Indivisibility):* A "basic" magma cannot be partitioned into two disjoint sub-magmas. (Proven via the properties of basic open sets).
3. M5 (Exhaustion):* Anything that is not a magma is either a standard set or an atom. (True by construction, as $M(A)$ is a subclass of the ZFA universe $V(A)$).

C. Set-Theoretic Properties of the Magmatic Universe

The paper analyzes which standard ZF axioms hold in the structure $\langle M^*, \in \rangle$ (where $M^* = M \cup A$):

• Powerset Axiom: Holds. The collection of sub-magmas of a magma $x$ forms a magma at the next level ($P(x) \cap M \in M_{\alpha+2}$).
• Union Axiom: Fails in general. The union of a set of magmas may result in a set of atoms (level 0), which is not a magma. The union axiom holds only for specific subsets (those that do not "touch" the bottom level of atoms).
• Other Axioms: Extensionality and Foundation hold. Empty Set, Pairing, and Infinity fail (as all magmas are infinite and non-empty).

D. Structural Insights

• Disjoint Levels: For finite $n \neq m$, the levels $M_n$ and $M_m$ are disjoint.
• Hierarchy Depth: The hierarchy never reaches a fixed point; $M_{\alpha+1} \neq M_\alpha$ for any $\alpha$.
• Transitivity: The class $M$ is not transitive (elements of $M_1$ are atoms, not sets in $M$), but $M \cup A$ is transitive.

4. Significance

• Bridging Philosophy and Logic: The paper provides the first rigorous mathematical model for Castoriadis' "magma," moving the concept from vague philosophical metaphor to a concrete set-theoretic structure.
• Challenging Independence: It demonstrates that set theory can accommodate "dependent" objects without abandoning the logical rigor of classical logic, provided one modifies the axioms regarding the nature of atoms and the hierarchy of sets.
• New Mathematical Objects: It introduces a new class of sets (magmas) where the standard intuition of "separability" is replaced by "connectivity" and "inseparability."
• Future Directions: The author suggests using magmas to create "magmatic codes" for standard mathematical objects (like natural numbers or functions) and exploring alternative axiomatic systems (referencing Skala's work) that inherently handle dependent elements.

Conclusion

Tzouvaras successfully constructs a "magmatic universe" where elements are inherently dependent on one another. By utilizing a pre-ordered set of atoms and a lower topology, he creates a hierarchy of sets that satisfies Castoriadis' core intuition: that certain collections cannot be broken down into independent parts. While the resulting structure does not satisfy all ZF axioms (notably Union and Pairing), it satisfies the specific axioms required to model the "magma" concept, offering a viable formal alternative to the standard Cantorian view of collections.