Excluded Middle as Computational Boundary

In the paper's words

This research investigates the validity of the constructivist critique of Turing completeness, which is founded upon the rejection of the Principle of the Excluded Middle (PEM). By employing a systematic, multi-layered framework, this paper deconstructs the logical chain from L.E.J. Brouwer’s intuitionistic philosophy to the perceived inadequacy of classical Turing/von Neumann architectures.

The abstract, verbatim; every line below is quoted from the text.

What it claims

This standard is intuitive, powerful, and forms the basis of

Numbers it reports

1. 1.3.7: Transition to the Constructivist Critique The deep integration of classical logic into the very hardware of modern computers creates a powerful and efficient paradigm for solving a vast range of problems.

   from 1.3:

2. 3.2.6: The Cost of Simplification: Discarded Information However, this powerful simplification comes at a cost: the loss of information.

   from 3.2: The Bit as

3. 4.5.5: The Thermodynamic Cost of Forced Certainty This destruction of information has a real physical cost, which can be understood in terms of thermodynamics.

   from 4.5:

4. 4.6.4: The Limits of Simulation Fidelity While a classical computer can simulate a complex system like the weather, the simulation is always an approximation.

   from 4.6:

5. 5.2.2: The Exponential Cost of Simulating “Becoming” The primary source of this inefficiency is the need to track a growing tree of possibilities.

   from 5.2:

6. 5.2.4: The Loss of “Algorithmic Elegance” Beyond the raw computational cost, the simulation lacks what could be called “algorithmic elegance.” The intuitionistic framework, with concepts like the BHK interpretation, provides a direct and natural mapping between logic and computation.

   from 5.2:

How it is done

1. 1.1.2: Enabling Proof by Contradiction One of the most powerful tools in the classical logician’s arsenal is the method of proof by contradiction, or reductio ad absurdum , a technique whose validity is directly guaranteed by the Principle of the Excluded Middle.

   from 1.1:

2. For an intuitionist, a mathematical object exists only when a finite method for its construction has been articulated.

   from 1.2:

3. Despite its simplicity, this model is believed to be capable of simulating any algorithm, a concept encapsulated in the Church-Turing Thesis.

   from 1.3:

4. A mathematical object or the solution to a problem does not exist until it has been explicitly constructed by a finite, verifiable procedure.

   from 1.5:

What would falsify it

1. To assert $P \lor \neg P$ constructively, one would need a finite method to either prove P for all numbers or find a counterexample to prove $\neg P$.

   from 1.2:

2. This would mean that Turing machines, and by extension all digital computers, contain a fundamental logical flaw that leads to incorrect or contradictory results.

   from 1.4:

3. Arguments that begin with a valid philosophical critique of PEM’s universal applicability frequently slide into claims that Turing machines are therefore “flawed” or “inadequate.” This conflation is a category error, mistaking a challenge to a system’s philosophical assumptions for a refutation of its internal, operational consistency.

   from 1.4:

Where the claim stops

1. Inquiry 7.6.1: The Theoretical Nature of the Argument The first and most significant limitation of this manuscript is its purely theoretical nature.

   from 7.6: Limitations of This

2. The goal, however, was not to provide a definitive account of any one field, but to build the bridges between them.

   from 7.6: Limitations of This

3. Our argument is that the constructivist critique provides a powerful theoretical motivation for pursuing these alternatives, but it does not prove that they will be successful.

   from 7.6: Limitations of This

Where to read

1. A198 words · 1 min
2. Chapter 1:5,858 words · 25 min
3. Chapter5,785 words · 25 min
4. Chapter5,524 words · 24 min
5. Chapter5,350 words · 23 min
6. Chapter5,297 words · 23 min
7. Chapter5,081 words · 22 min
8. Chapter5,138 words · 22 min
9. Appendices401 words · 2 min