Abstract

Starting from Turing’s 1939 notions of Ordinal Logic and the Oracle Turing Machine, and incorporating Penrose’s reinterpretation of the “Cautious Oracle,” this paper delves into the tension between formal systems and human cognition. We highlight how Turing dissected mathematical reasoning into two faculties: ingenuity (rule-based deduction) and intuition (non-mechanical judgment). Penrose further proposed the “Cautious Oracle” to model credibility in human judgment. By drawing an analogy between the act of counting and the mechanism of Bitcoin mining, this article reveals how ordinal logic serves as a bridge between uncomputable judgment and verifiable processes.

I. Turing’s Insight: The Split Between Ingenuity and Intuition

In his 1939 doctoral dissertation, Turing broke down mathematical reasoning into two essential faculties:

• Ingenuity: The capacity to reason using formal rules—modeled as step-by-step deductions akin to a Turing Machine.
• Intuition: A non-mechanical human ability to judge the truth of a statement without relying on a formal chain of reasoning.
  

Turing argued that a truly intelligent system cannot rely on ingenuity alone. Humans often step outside formal rule systems to make decisions such as “this can be accepted as a new axiom” or “this is an admissible construction.” These judgments require a higher-order cognitive ability—intuition—that cannot be derived from existing rules.

II. Ordinal Logic: Making Room for Intuition

In response to Gödel’s incompleteness theorems, Turing proposed Ordinal Logic:

• Formal systems are arranged according to ordinals \alpha, creating a hierarchy of increasingly powerful systems S_\alpha.
• Each level S_\alpha is built by identifying formulas \phi_\alpha suitable for extension from the previous level.
• The act of identifying such formulas cannot be performed by existing rules—it requires intuition.
  

In other words, each transition point in ordinal logic is a point of intuitive intervention.

This is where Turing’s depth shines: he didn’t reject formal systems but acknowledged that to transcend them, one must involve non-mechanical, external judgments.

III. Oracle Turing Machines: A Formal Expression of Intuition

To model judgments beyond the capabilities of standard Turing Machines, Turing introduced the Oracle Turing Machine:

• During its operation, it can query an “oracle” to ask a question unsolvable by conventional Turing machines (e.g., the halting problem).
• The oracle acts like a “black-box judge” that can directly answer “yes” or “no.”
• In theory, the oracle grants relative computability—allowing computation relative to an uncomputable set.
  

This concept is often misunderstood as a “universal machine,” but Turing’s intent was not to construct omnipotence. Rather, it was to allow formal systems to assume a form of judgment in order to build more complex systems.

IV. Penrose and the Cautious Oracle: Making Trust Discussable

In The Once and Future Turing, Penrose reinterpreted the idea of the oracle and proposed a model more aligned with human behavioral patterns—the Cautious Oracle:

• The Cautious Oracle does not always give an answer.
• When faced with uncertainty, it may remain silent or enter a state of “continued effort.”
• Once it outputs an answer, that answer must be trustworthy and true.
  

Penrose defined three possible behaviors of the oracle:

• \mathcal{D}(g) = \text{true}
• \mathcal{D}(g) = \text{false}
• \mathcal{D}(g) = ?, or enters an infinite “loop” searching for the answer
  

This “waitable yet trustworthy” mechanism models the cautious strategies of human experts in complex judgments and resonates with the scientific attitude of “no conclusion yet.” It relies not on omniscience, but on continuous effort and the establishment of limited trust.

V. Bitcoin Miners: The Real-World Cautious Oracle

In the Bitcoin system, miners play a role remarkably similar to the Cautious Oracle:

• Miners evaluate whether each transaction is valid (e.g., free from double spending).
• They engage in Proof of Work—continuous effort to find an acceptable solution.
• Once a solution is found, it is broadcast to the network and verified/trusted by other nodes.
  

This is a consensus model not bound to internal formal systems:

Bitcoin miners are real-world embodiments of the Cautious Oracle—fallible but trustworthy, imperfect but stable. They represent a distributed engineering implementation of non-formal judgment.

VI. Intuition Exemplified in the Act of “Counting”

This kind of “intuitive judgment” is not merely philosophical. Consider this common example:

When we say “count the fifth apple,” it appears mechanical—but actually involves non-mechanical judgment: we must first determine “this is an apple” before including it in the count.

That judgment does not arise from strict definitions but from experiential perception and conceptual induction. This intuitive recognition of “same-category” objects is precisely the aspect Turing believed could not be mechanized.

Turing had already realized that formal systems cannot cover the entire source of real-world judgments. His construction of ordinal logic and oracles was an attempt to build a bridge for human judgment within a world of pure rules.

VII. Conclusion: A Path from Uncomputability to Verifiable Trust

Turing, Penrose, and the Bitcoin system—through logic, cognitive science, and engineering—depict a shared structure:

• We do not deny incompleteness and uncomputability.
• But we can build mechanisms that balance judgment between trust and verifiability.
• The core of this mechanism is granting intuition space within formal models, introducing non-formal judgment via external devices (oracles, miners).
• Ultimately, this establishes a judgment pathway that transcends rule systems—not absolute truth, but trustworthy truth.
  

This is the deepest contemporary echo of Turing’s thought.