On Computable Numbers and Computing Machinery

Summary

This paired entry brings together Alan Turing's 1936 paper On Computable Numbers, with an Application to the Entscheidungsproblem and his 1950 essay Computing Machinery and Intelligence. The first paper attacks a precise mathematical problem: what does it mean for a number or function to be computable? Turing answers by imagining a simple abstract machine that reads and writes symbols on a tape according to a finite table of instructions. By stripping calculation down to discrete operations, memory, and rule-governed motion, he turns human computation into a formal object. The paper proves that there is no general method for deciding every mathematical proposition of the relevant kind, linking Turing's work to Hilbert's program and Godel's incompleteness results. It also gives later engineers and theorists a way to compare many different machines under one concept of effective procedure. The later essay changes tone and audience. Rather than asking what can be computed, it asks whether machines can think, then replaces the vague question with an operational test involving conversation and imitation. Turing surveys objections from theology, consciousness, disability, Lady Lovelace, mathematics, nervous systems, and extrasensory perception, responding with a mixture of logical discipline and speculative boldness. He does not claim that a machine has a soul or a private inner life; instead, he asks what evidence would count in ordinary judgment. Together the two works define computing as both formal limitation and cultural provocation. Turing shows that mechanical procedure has exact power, then asks how far such procedure may enter territories once reserved for mind. The pair is therefore foundational not only for computer science but for psychiatry, cognitive science, and philosophy of mind. It makes intelligence discussable as behavior, symbol manipulation, learning, error, and social judgment, while never letting readers forget that the most powerful machines are bounded by mathematical impossibility.

Plot or Argument

• Computation becomes a machine

  The 1936 paper defines computability through an idealized device with a tape, symbols, states, and rules, transforming informal calculation into a formal model.

• The decision problem fails

  Turing applies the model to show that no universal decision procedure can settle all cases of the relevant mathematical problem.

• The thinking question is displaced

  The 1950 essay reframes whether machines can think as an imitation-game question about performance, conversation, and judgment.

• Objections become a map of anxiety

  Turing's replies to theological, conscious, mathematical, neurological, and Lovelacean objections show where mid-century culture located the boundary around mind.

Lessons

• Definition can create a field

  Turing's machine is powerful because it defines computation in a form that can be reasoned about. The lesson is that a good abstraction does not merely simplify a problem; it makes new proofs, machines, and disciplines possible.

  • Why is Turing's stripped-down machine more useful than a realistic description of a calculator?
  • How does the abstraction handle memory and instruction?
  • What modern fields were made possible by a similarly precise definition?

• Limits are discoveries, not defeats

  The undecidability result is negative, but it gives computation a rigorous boundary. Turing shows that knowing what cannot be done is part of understanding what mechanical procedure is.

  • Why does an impossibility proof count as progress?
  • How does Turing's limit relate to Godel's incompleteness theorem?
  • Where do contemporary systems still hide impossible expectations?

• Mind can be approached through behavior

  The imitation game moves debate from inner essence to observable exchange. This does not settle consciousness, but it gives a practical way to test claims about intelligence without pretending to inspect another mind directly.

  • What does the imitation game avoid by refusing to define thinking?
  • Does behavioral success prove intelligence or only persuade observers?
  • How does this matter for clinical interviews and artificial agents?

Context

The 1936 paper belongs to mathematical logic after Hilbert and Godel, when the limits of formal systems became urgent. The 1950 essay appears in the early electronic-computer era, after wartime codebreaking and before artificial intelligence existed as a named discipline.

Composition

On Computable Numbers was published in 1936 with a correction in 1937. Computing Machinery and Intelligence appeared in Mind in 1950, drawing on Turing's postwar computer work and his willingness to address public philosophical objections.

Reception

The machine model became central to computability theory, while the imitation game became the best-known early test of machine intelligence. Later AI debates often return to Turing either as origin, foil, or unresolved challenge.

Comparisons

• notes-on-menabreas-sketch-of-the-analytical-engine

  Turing explicitly engages the Lovelace objection and shifts the issue from mechanical execution to learning, surprise, and conversational performance.

• formally-undecidable-propositions

  Godel reveals limits within formal arithmetic; Turing recasts related limits through effective procedure and an abstract machine.

• logical-calculus-of-ideas-immanent-in-nervous-activity

  McCulloch and Pitts model neurons logically, while Turing models computation procedurally and later asks how machines might imitate intelligent behavior.

• laws-of-thought

  Boole formalizes logical relations; Turing formalizes the act of calculation itself and asks whether such formal action can bear the name intelligence.

Quotes

For Students

A reading companion for classical Christian high school students. See the student guide.

Reading Guide

Watch for

• In the 1936 paper, the moment Turing justifies his machine by imagining a human clerk computing with pencil and squared paper, able to look at only a few symbols at once and to be in only finitely many states of mind. The machine is built out of a description of a person doing arithmetic.
• The universal machine of the 1936 paper: one machine that, given the written description of any other machine on its tape, can do whatever that machine does. Notice that instructions have become data.
• The diagonal argument near the end of the 1936 paper, where Turing shows that no machine can decide, for every machine description, whether that machine will go on printing digits. This is the hinge that settles the Entscheidungsproblem in the negative.
• The opening of the 1950 essay, where Turing sets aside the question whether machines can think as too vague and replaces it with the imitation game played by an interrogator, a person, and a machine.
• The long section answering objections in the 1950 essay, especially the theological objection, the argument from consciousness, and Lady Lovelace's objection, followed by the closing proposal to build a child machine and teach it.

Key terms and names

Entscheidungsproblem

German for decision problem: David Hilbert's challenge to find a mechanical method that could decide whether any given statement of first-order logic is provable. Turing's 1936 paper shows that no such method exists.

Computing machine (Turing machine)

Turing's idealized device with an unbounded tape divided into squares, a head that reads and writes one symbol at a time, and a finite table of states and rules. Anything a human clerk could compute by rule, Turing argues, such a machine can compute.

Universal machine

A single machine that imitates any other computing machine when given that machine's description on its tape. It is the theoretical ancestor of the stored-program computer.

Computable number

A real number whose decimal expansion a machine can write out digit by digit by a finite set of rules. Turing shows that the computable numbers are countable, so most real numbers are not computable.

Imitation game

The test proposed in the 1950 essay: an interrogator questions two hidden players by typed messages and must decide which is the machine. Turing asks whether a machine could play well enough to make the interrogator's guess no better than chance.

Lady Lovelace's objection

Ada Lovelace's remark in her 1843 notes that the Analytical Engine can originate nothing and can do only what we order it to perform. Turing replies that machines can take people by surprise and that learning may let a machine go beyond what its makers foresaw.

Child machine

Turing's proposal at the close of the 1950 essay to build a simple machine resembling a child's mind and educate it, rather than trying to program an adult mind directly.

Seminar Questions

1. Comprehension What parts make up Turing's computing machine in the 1936 paper, and why does Turing think a human clerk computing with paper can do nothing the machine cannot?
2. Comprehension How is the imitation game played in the 1950 essay, and what question does Turing put in place of the question whether machines can think?
3. Comprehension What is Lady Lovelace's objection, and how does Turing answer it?
4. Interpretation The 1936 paper proves that some questions about machines can never be settled by any machine. Why does a result about the limits of computation open a field rather than close one?
5. Interpretation In the 1950 essay Turing answers the theological objection by saying that it would restrict God's power to deny that He could give a soul to a machine if He chose. Does this reply take the objection seriously, or change the subject?
6. Interpretation Turing says the argument from consciousness, pressed to its end, leads to the view that one can only be sure one thinks oneself. Is he right that we judge other people's minds by their behavior, and does that justify judging machines the same way?
7. Big Question If a machine could answer questions in conversation so well that no interrogator could tell it from a person, would that show it understands, or only that understanding cannot be seen from outside?

Faith Connections

• Genesis 2:7 (Scripture)

  The Lord God forms man from the dust and breathes into him the breath of life, so that he becomes a living soul. Turing's theological objection turns on exactly this question of whether life and mind are given or can be built, and Genesis places the answer in an act of God rather than in the arrangement of matter alone.

• Psalm 139:13-16 (Scripture)

  The psalmist praises God for being fearfully and wonderfully made, known before birth. Set beside the child machine, the psalm suggests that the difference between a person and an educated machine may lie less in performance than in being known and loved by a Maker.

• René Descartes, Discourse on Method, Part V (Great Conversation)

  Descartes argued that a machine might imitate many bodily actions but could never arrange words to answer sensibly to whatever is said in its presence, as even a dull person can. Turing's imitation game takes up almost exactly this test and bets the other way.

Logic and Rhetoric

Central argument: In 1936: whatever a human can compute by following rules can be done by a simple machine reading and writing symbols; such machines can be listed and described as numbers; a diagonal argument then shows that no machine can decide for every machine whether it will keep printing; therefore no mechanical procedure solves Hilbert's decision problem. In 1950: we cannot inspect any mind but our own and judge others by what they say and do; a machine that converses indistinguishably from a person gives the same evidence; therefore the question whether machines can think should be replaced by whether they can play the imitation game, and no stated objection shows that they cannot.

Rhetorical device: procatalepsis. The 1950 essay devotes its longest section to stating nine objections against its own view, from the theological objection to the argument from extrasensory perception, in their strongest form before answering each. By naming the reader's resistance first, Turing makes his own position look like what remains standing.

Debate: Resolved: A machine that reliably passes Turing's imitation game should be said to think.

Writing Prompts

Essay: Turing's 1936 paper defines computation by imagining a person following rules, and his 1950 essay defines thinking by what a person can observe in conversation. Argue whether these two moves are the same kind of method, and whether that method is adequate to the question of what a mind is.

Comparison with Notes on Menabrea's Sketch of the Analytical Engine: Ada Lovelace wrote that the Analytical Engine can only do what we know how to order it to perform; a century later Turing answered her by name. Compare Lovelace's account of what a calculating engine can do with Turing's account of learning machines, and argue which of them better describes the limits of a machine.

Medical Lens

Turing is central for psychiatry-adjacent readers because the 1950 essay recasts mind as behavior, learning, communication, and judgment by observers. That frame touches cognitive assessment, artificial intelligence in clinical settings, and the ethics of attributing agency to machines. The essay's discussion of nervous systems and disability also records mid-century assumptions about embodiment, cognition, and social evaluation.

• 1936 paper, definition of computing machines

  The reduction of calculation to symbolic operations offers a formal ancestor for computational models of cognition.

  Mind BodyPsychology Unconscious
• 1950 essay, imitation game

  The replacement of the thinking question by a behavioral test matters for psychiatric interviews and modern AI evaluation.

  Medical Ethics
• 1950 essay, objections about nervous systems and disability

  Turing's survey of objections shows how embodiment and impairment enter debates about intelligence.

  Disability DeformityMind Body

Creative Prompts

• Essay

  Compare the Turing machine and the imitation game as two different ways of refusing vague definitions.

• Dialogue

  Write a consultation between Turing and a psychiatrist about whether conversational fluency is evidence of understanding.

Further Reading

• Alan Turing: The Enigma, Andrew Hodges

  Major biography connecting Turing's mathematics, war work, sexuality, prosecution, and death.

• The Essential Turing, B. Jack Copeland

  Edited selection of Turing's writings with commentary.

• Godel, Escher, Bach, Douglas R. Hofstadter

  Influential meditation on formal systems, minds, and machines.