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That the human mind has no such limits has only been stated, without any sort of proof. We give wrong answers too often ourselves to be pleased at the machines’ fallibility. And there may be people cleverer than any given machine, but then there may be other machines cleverer again.

The mathematical objection: results of mathematical logic, such as Gödel’s theorem, show that every machine has limits. There are questions it will answer wrongly, or not at all, however long it’s given. The human mind has no such limits.

That’s a hope, not an argument, and it isn’t substantial enough to require refutation. Consolation would be more appropriate.

The heads-in-the-sand objection: the consequences of machines thinking would be too dreadful. Let us hope and believe that they can’t.

That restricts the omnipotence of the Almighty. Couldn’t He confer a soul on an elephant, or on a machine, if He saw fit? Besides, theological arguments have often failed before: in Galileo’s time, Bible verses were held to refute Copernicus.

The theological objection: thinking is a function of man’s immortal soul. God has given an immortal soul to every man and woman, but not to any other animal or to machines. So no animal or machine can think.

Machines take me by surprise with great frequency. To think that nothing new can come from following given rules is to assume that as soon as we know a fact, all its consequences spring to mind at once. They don’t: working them out is real work.

A machine can be made to learn. Instead of programming it to act like an adult mind, program it like a child’s, and educate it. Its teachers won’t know quite what is going on inside it, so it won’t only do what we know how to order it to do.

Machines will be able to think. The question “Can machines think?” is too meaningless to deserve discussion as it stands. Ask instead whether a machine can play the imitation game, passing for a person in typed conversation. In about fifty years, computers will play it so well that an average interrogator won’t have more than a 70 percent chance of telling them apart after five minutes of questioning, and one will be able to speak of machines thinking without expecting to be contradicted.

Whatever a person can compute by following rules, a simple machine can compute, reading and writing symbols on a tape. And a single machine, given a description of any other on its tape, can do whatever that machine does: one universal machine can compute anything that can be computed at all.

No machine can decide, for every statement of logic, whether it can be proved. If one could, it could decide whether any machine ever prints a 0, and a machine that asked it that about itself could be made to do the opposite.

Some problems can’t be solved by any effective calculation: I’ve found one in elementary number theory. So no mechanical procedure can decide, for every statement of logic, whether it can be proved.

Truth for a language rich enough for arithmetic can be defined only in a richer language, never within the language itself. So no one system of symbols and rules can even say which of its own statements are true: there’s no universal criterion of truth.

In any consistent system of rules strong enough for arithmetic, there are true statements that the rules can neither prove nor disprove. So no one system can answer every question of mathematics.

Every question of mathematics can be answered: there is no “we will not know.” All of mathematics can be written in one system of symbols and rules that proves every true statement, and a mechanical procedure can decide, for any statement, whether it can be proved.

The Analytical Engine can’t originate anything. It can do whatever we know how to order it to perform, and follow our reasoning, but it can’t anticipate any truths we haven’t worked out ourselves.

The Analytical Engine needn’t stop at numbers. It works on symbols by rules, and symbols can stand for anything whose relations can be put into rules, like the notes of music. So it could compose music, or work in any field whose rules we can write down.

A machine can carry out any calculation at all, however long. Punched cards tell it which operations to perform on which numbers. It keeps the numbers in a store, works on them in a mill, and chooses which cards to follow next by its own results.

Mathematical tables can be worked out and printed by machinery. Almost any table can be built up by adding differences, and gears can add, so a machine of number wheels can do it, without the mistakes that human computers and typesetters make.

Write every idea in a universal language of symbols, and fixed rules for calculating with them, which a machine could follow, will settle any question that doesn’t turn on facts about the world. Then such disputes can be settled without arguing: the two sides can sit down and calculate.