The Intuitionistic Continuum

Abstract

The intuitionistic continuum is the dynamic process of generating mathematical objects over time, rather than a static set of completed points. It relies on the human cognitive ability to distinguish discrete moments (the passage from “now” to “next”), which allows us to construct sequences. This structure serves as the underlying “stream” or “generator” from which specific real numbers (defined as infinite approximations) are produced.

It is not just the reals

1. Classical View: “The Continuum” and “The Set of Real Numbers” are synonymous. Both are static sets of points.
2. Intuitionistic View:
   • The Continuum is the primitive, viscous medium. It is “indecomposable” (cannot be cut into two disjoint sets).
   • The Reals ($\mathbb{R}$) are the species (collection) of explicitly constructed points (Choice Sequences) residing in this medium.
   • The Continuum acts as the generator; $\mathbb{R}$ is the generated content.

Key Properties

• Non-Atomistic: It is not composed of points; points are constructed on it.
• Viscosity: It “sticks together.” The theorem “All functions $f:\mathbb{R}\to \mathbb{R}$ are continuous” holds because the domain is not a collection of discrete dust that can be mapped discontinuously.

References

1. Primordial Intuition of Time

Brouwer’s foundational axiom. It describes the fundamental mental act of distinguishing two separate moments: a “past” moment and a “present” moment.

• Philosophy: The “falling apart of a life-moment into a two-ity.”
• CS/Type Theory: This is the cognitive basis for Inductive Types. It is the recognition of discrete steps, allowing us to construct the Natural Numbers ($\mathbb{N}$) via Zero and Succ. It grounds arithmetic in the human capacity for sequential processing.

2. Medium of Free Becoming

The view that the continuum is not a static “being” (a completed object) but a dynamic process of “becoming.”

• Philosophy: Points on the continuum are never finished; they are always “in progress.”
• CS/Type Theory: This corresponds to Coinduction or Streams. A real number is a Stream Limit produced by a generator. It is “free” because the generator is not restricted to a fixed algorithm—it can accept external inputs (like sensor data or user choices) indefinitely.

3. Viscosity (Indecomposability)

The property that the continuum “sticks together” and cannot be split into disjoint parts.

• Philosophy: The continuum is not a pile of sand (atoms); it is a fluid. If you try to cut it, the cut itself is a construction that requires time, preventing a clean, instantaneous separation.
• CS/Type Theory: This is Topological Connectedness.
  • Classical: $X$ is connected if $X\ne U\cup V$ for disjoint open sets.
  • Constructive (Stronger): Any continuous function $f:\mathbb{R}\to \text{Bool}$ must be constant. You cannot output true for the left half and false for the right half, because the decision boundary requires infinite precision to resolve.

4. Species

Brouwer’s term for what we typically call “sets” or “subsets” defined by a property.

• Philosophy: Since “sets” were often associated with completed infinite collections (which Brouwer rejected), he used “Species” to denote a collection defined by a property.
• CS/Type Theory: A Predicate or Dependent Pair. A species $S$ on a type $A$ is defined by a proposition $P:A\to \text{Prop}$. The elements are pairs $(x,p)$ where $p$ is a proof of $P(x)$.

References

• vanatt2003-choice
• brouwer1981-lectures
• taylor1999-practical-foundations