Reconstructive Ontology of Brouwerian Intuitionism

In the paper's words

This study reconstructs the ontological foundations of Brouwerian intuitionism, arguing that the rejection of the principle of the excluded middle (PEM) is not a technical choice but a necessary consequence of the ur-intuition of time. Brouwer’s primary texts from 1905 to 1948, we demonstrate how the creating subject serves as the sole architect of mathematical truth, positioning language as a secondary, often deceptive, “lifeless superstructure” (Brouwer, 1905). We employ a dual methodological approach, combining genealogical textual analysis with a computational simulation of drifts (choice sequences) to operationalize the subject’s temporal constraints.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. However, our computational model returned “Undecided” for these micro-drifts because the deviation fell below the verification threshold ($\epsilon = 10^{-9}$).

   from 4.7 Computational Simulation

How it is done

1. We employ a dual methodological approach, combining genealogical textual analysis with a computational simulation of drifts (choice sequences) to operationalize the subject’s temporal constraints.

   from Abstract

2. The turning into oneself advocated in 1905 became the method of introspective construction in 1907, where the subject retreats from the external world to observe the falling apart of time.

   from 1.2 Historical Context:

3. Reconciling the private nature of the ur-intuition with the public necessity of mathematical proof remains the central tension of the intuitionistic project.

   from 1.4 The Problem of Language

4. This reconstruction addresses the paradox of communication by establishing that intuitionistic proof is a normative proposal for intersubjective synchronization, grounded in the shared human capacity for temporal experience (Dummett, 2000).

   from 1.5 Thesis Statement

What would falsify it

1. If the two-ity is merely a psychological event, then mathematics would seem to lack the universal necessity required of a rigorous science.

   from 3.1 The Falling Apart of the

2. In the context of the drift, let $P$ be the proposition “A checking-number $k$ exists.” The negation $\neg P$ would mean “It is absurd that $k$ exists” (i.e., we can prove $k$ will never occur).

   from 4.4 Double Negation and

3. Testability Brouwer generalized these findings into the simple principle of testability, which asserts that any meaningful mathematical proposition must be tested against the subject’s construction capability (Brouwer, 1948).

   from 4.5 The Simple Principle of

Where the claim stops

1. However, this foundational text was heavily sanitized at the behest of his advisor, Korteweg, who urged Brouwer to suppress the mystical and solipsistic elements present in his earlier 1905 manifesto, Life, Art, and Mysticism (Brouwer, 1907).

   from 1.2 Historical Context:

2. The trajectory of this derivation is strictly unilateral: logic and language do not precede mathematics; rather, mathematics flows directly from the temporal activity of the subject.

   from 1.3 The Ur-Intuition Defined

3. The ur-intuition thus serves as both the genesis of number and the boundary condition for truth—nothing can be true that cannot be constructed within this temporal flow.

   from 1.3 The Ur-Intuition Defined

Where to read

1. Abstract170 words · 1 min
2. Keywords15 words · 1 min
3. 1.0 Introduction: The1,316 words · 6 min
4. 2.01,148 words · 5 min
5. 3.0 Results I: The3,633 words · 16 min
6. 4.0 Results II: The1,317 words · 6 min
7. 5.0 Discussion: The Paradox963 words · 4 min
8. 6.0 Conclusion: The Subject’s780 words · 3 min
9. Appendices372 words · 2 min