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Turing 1936 — On Computable Numbers

Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem." Proceedings of the London Mathematical Society, Series 2, Vol. 42, pp. 230-265. DOI: 10.1112/plms/s2-42.1.230. Some problems cannot be solved by

The Source

Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem." Proceedings of the London Mathematical Society, Series 2, Vol. 42, pp. 230-265. DOI: 10.1112/plms/s2-42.1.230.

The Claim

Some problems cannot be solved by any mechanical procedure. [SOURCE:turing-1936|type:mathematical] Turing proved this by inventing a machine that defines what "mechanical" means.

The Context

Hilbert asked for a decision procedure. He wanted a single algorithm that could settle every mathematical question. Mathematicians believed such a procedure existed. They were wrong. Turing was twenty-four. He solved the problem by imagining a machine.

The year was 1936. Europe was darkening. Gödel had already shattered completeness in 1931. [SOURCE:godel-1931|type:mathematical] The foundations of mathematics were in crisis. Hilbert's program was the last hope: a mechanical procedure to decide all truths. Turing ended that hope with a thought experiment.

The Evidence

Turing defined a computable number as one whose decimal digits a machine could print. The machine reads a tape. It moves left or right. It writes symbols or erases them. Its behavior is determined by a finite table of instructions. This is the Turing machine. [SOURCE:turing-1936|type:mathematical]

Turing then constructed a universal machine. One machine that can simulate any other. Feed it the description of any Turing machine and its input. It computes what that machine would compute. [SOURCE:turing-1936|type:mathematical]

Then he proved the halting problem. No machine can predict whether another machine will halt or run forever. The proof is a diagonal argument. The machine is asked to judge itself. Contradiction follows. Therefore no such machine exists. [SOURCE:turing-1936|type:mathematical]

The Entscheidungsproblem falls immediately. If you cannot determine whether a machine halts, you cannot determine whether a theorem is provable. The limit is absolute.

The Convergence

This source instantiates C20 — Universal Computation. [SOURCE:turing-1936|type:mathematical]

Turing's machine is the abstract structure that underlies every computer. One machine simulates all others. This is not metaphor. It is theorem.

The paper also instantiates C08 — Recursion / Self-Reference. [SOURCE:turing-1936|type:mathematical] The diagonal argument requires a machine to examine its own behavior. Self-reference produces undecidability.

The convergence is triple. Church proved the same result independently, using lambda calculus. [SOURCE:church-1936|type:mathematical] Post arrived independently with finite combinatory processes. Three methods, one limit. The boundary of computation is real.

The Honest Limits

Turing assumed a discrete, deterministic machine. Nature is not discrete. Quantum mechanics is probabilistic. Whether the universe itself is computable remains open. [SOURCE:turing-1936|type:theoretical]

Turing did not address computational complexity. A problem can be computable yet take longer than the age of the universe to solve. P versus NP was decades away.

His machine has infinite tape. Real machines have finite memory. The idealization matters for some proofs.

The Church-Turing thesis is a hypothesis about physics, not a theorem. It may fail at quantum or biological scales. [SOURCE:turing-1936|type:theoretical]

The Receipt

"We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions... The machine is supplied with a 'tape' (the analogue of paper) running through it, and divided into sections (called 'squares') each capable of bearing a 'symbol'."

This is §1 of the paper. Turing constructs the machine from scratch.

The diagonal argument, in his own words:

"It follows that there can be no machine E which, when supplied with the S.D [standard description] of any computing machine M, will determine whether M ever prints a given symbol..."

The machine D, applied to itself, produces a contradiction. Therefore D cannot exist. The receipt is complete. [SOURCE:turing-1936|type:mathematical]

Related Sources

  • [SOURCE:church-1936|type:mathematical] — Alonzo Church proved the same undecidability independently via lambda calculus
  • [SOURCE:godel-1931|type:mathematical] — Gödel's incompleteness theorems set the stage for Turing's limit
  • [SOURCE:post-1936|type:mathematical] — Emil Post arrived independently with finite combinatory processes
  • [SOURCE:von-neumann-1945|type:empirical] — Von Neumann built the stored-program computer architecture from Turing's blueprint
  • [SOURCE:shannon-1948|type:theoretical] — Shannon's information theory completes the triad: computable, communicable, compressible
  • [SOURCE:prigogine-1977|type:empirical] — Dissipative structures push against the same limits: what can be computed versus what can exist
References (5)
  1. On Computable Numbers, with an Application to the Entscheidungsproblem (1936)
  2. On Formally Undecidable Propositions of Principia Mathematica and Related Systems (1931)
  3. A Mathematical Theory of Communication (1948)
  4. Irreversibility and Heat Generation in the Computing Process (1961)
  5. Self-Organization in Nonequilibrium Systems (1977)