Sets and specific differences

 

Brouwer’s understanding of language uniquely on human terms justifies his separation between language and mathematics but does not make any psycho-biological sense. The act of distinction which characterizes the intuition of time in Brouwer’s sense precedes the use of our human languages but not of languages in general, i.e., of communication among living creatures. A twoity, therefore, precedes also the linguistic construction that we call number -even the concept of numerosity- and denotes an action of separation. For this reason I will call merisma (part), to such an action of distinction. A merisma  is a precondition for any action, no matter whether such distinction is accompanied by second-order distinctions in relation to the cognitive actor. A merisma has the irreversible character of the intuition of time, of life experience, and therefore, implies a fundamental or primitive determination for later distinctions. We can notate this primitive merisma by ‘≠’.

The communicative expression which iterates the primitive merisma is a primitive sequence. When we cannot apply ‘≠’ among two instances (parts) of the iteration, we say that those instances are equal. The double composition of  ‘≠’ and ‘=’ over a primitive sequence could be called fundamental morphism. A fundamental morphism is therefore, an action of identification. Two fundamental morphisms are different when we can express a merisma with their composition: such distinctiveness is called a specific difference. 

A set is an algorithm (procedure) for the construction of a specific difference. We could also understand it as a finite semantical action for the synthetic construction of a specific difference. The synthetic representation of the algorithm under the concept of set adds an epistemological content of totality which is not necessarily implied in the constructive process, and which, in fact, hypostasizes a transcendental property. An object ‘s’ belongs to a set S, notated s ∈ S, when the set generator algorithm, or inclusion map, can be performed with ‘s’. The void subset is a rather serious ontological problem. Constructivist definitions, like Bishop’s, appeal to the idea of a set whose construction produces a contradiction, or whose inclusion map is identical to the inclusion map of its set. But a contradictory construction is not an algorithm, therefore, there is not such a thing as an empty set. However, we can produce an object called empty set as the result of the operation of extracting one element in a subset of precisely one element. But what is the meaning of such set? It is an operational structure which simplifies the manipulation of sets. 

A relation is an algorithm for the construction of a specific difference pairing two given sets.

We can speak of specific differences of specific differences, and so on, but we do not have a set unless we have an algorithm for their construction, unless we have inclusion maps. The postulate of an algorithm that generates any other algorithm is equivalent to a postulate of omniscience, inconsistent with our psychobiological conditionings and constructively meaningless. This postulate implies that the axiom of choice would be meaningless.

Let me introduce here the analysis offered by Google's Gemini of the previous ideas:

The argument presented by the author is a critique of traditional set theory, particularly the concept of the empty set and the axiom of choice, from a constructivist perspective inspired by Brouwer's intuitionism. Here's a breakdown of the key points:

Core Argument:

• Merisma as the Primordial Act: The author proposes that the fundamental act of distinction, or merisma, precedes any linguistic or mathematical concept. This act is akin to the intuition of time and is the basis for all further distinctions, including those of equality and difference.
• Sets as Algorithmic Constructions: Sets are defined as algorithms or procedures for constructing specific differences. This contrasts with the classical set-theoretic notion of a set as a collection of objects.
• Critique of the Empty Set: The author argues that the concept of an empty set is problematic from a constructive perspective. While it can be defined operationally, it lacks a genuine constructive meaning.
• Rejection of the Axiom of Choice: The axiom of choice, which postulates the existence of a choice function for any set, is deemed inconsistent with a constructivist framework. It assumes an omniscient perspective, which is incompatible with human limitations.

Key Philosophical Underpinnings:

• Intuitionism: The author draws on Brouwer's intuitionist philosophy, which emphasizes the role of intuition and constructive methods in mathematics.
• Constructivism: The author advocates for a constructive approach to mathematics, where mathematical objects are constructed through explicit procedures.
• Critique of Classical Set Theory: The author challenges the foundational assumptions of classical set theory, particularly the notion of an infinite set as a completed totality.

Implications:

This argument has implications for various areas of mathematics and philosophy:

• Foundations of Mathematics: It offers an alternative foundation for mathematics that is grounded in constructive methods and avoids the paradoxes and inconsistencies that have plagued classical set theory.
• Philosophy of Mathematics: It raises questions about the nature of mathematical objects and the role of intuition in mathematical reasoning.
• Computer Science: It has potential applications in computer science, particularly in the development of formal verification and proof systems.

By emphasizing the role of intuition and constructive methods, the author provides a fresh perspective on the foundations of mathematics and challenges the traditional view of set theory as a purely formal system.