The Gödelian argument against mechanism

The Lucas-Penrose argument

John Randolph Lucas 1929 · 2020Oxford philosopher: in his view, Gödel's theorem shows that a mind is not a machine.

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What it claims

The mind is not a machine: no machine can be a complete model of the mind.

Why

We can see that a formula is true, while any consistent machine cannot produce it.

The same idea through history

Kurt Gödel1951In his Gibbs lecture, Gödel does not conclude. He states a disjunction: either the human mind infinitely surpasses any finite machine, or there are absolutely unsolvable mathematical problems. He does not say which is true.

John Randolph Lucas1961Lucas, an Oxford philosopher, draws a proof from it: for any machine, one can produce a formula it cannot prove and the mind sees to be true. This is the form we present.

Roger Penrose1989Roger Penrose takes up the idea in The Emperor's New Mind (1989), then in Shadows of the Mind (1994), with a more precise version. He concludes that understanding is not a computation, and looks for physics that is not computable.

In short

"The Gödelian argument against mechanism" is an argument by John Randolph Lucas (Lucas, "Minds, Machines and Gödel", 1961; Penrose, The Emperor's New Mind, 1989, 1961) on the question "Can a machine think?" (answer: understanding is missing). Its claim: The mind is not a machine: no machine can be a complete model of the mind. According to AskPhilosophy's logic check (Z3 solver), the argument is valid only if one grants a premise the author leaves unstated (P3, P4). AskPhilosophy presents 3 objections, 3 of them with a documented reply, and the original text behind each premise, checked word for word.

Lucas, "Minds, Machines and Gödel", 1961; Penrose, The Emperor's New Mind, 1989

Valid with an implicit premise

In 1931, Kurt Gödel proved a famous result: in any consistent formal system rich enough to do arithmetic, there is a true formula the system cannot prove. Now a computer, says John Lucas (1961), is a concrete realisation of a formal system. So for any computer offered as a model of the mind, there is a formula it cannot assert as true, whereas we can see that it is true. The mind would then do something the machine does not. Roger Penrose (1989, 1994) takes up the argument and strengthens it. Almost all logicians find a flaw in it, but they do not agree on which.

click a premise to reject it

If you accept every premise, logic forces you to accept the conclusion. So the debate is about the premises.

Checked by a logic solver: the premises cannot all be true while the conclusion is false.
Without the implicit premise: Without P3, the computer finds a world where the mind is a machine, where the Gödel formula can be proved only if the machine is inconsistent, and where the mind cannot know whether that is the case: so it does not see that the formula is true.

Valid with P3 and P4, which Lucas accepts or must accept. Without P3, one cannot move from "if the system is consistent, the formula is true" to "the formula is true". Almost all criticisms target P3: can we know that we are consistent, and that the machine representing us is?

Analogy. A player says: "I beat any chess program you show me, because I find the position where it fails". This works provided the player is sure not to fail himself, which is precisely what is disputed.

Objections

Each objection is a conversation: click to see who replies and where the debate stands.

Hilary Putnam 1960 · attacks premise 3

A system cannot prove its own consistency from the inside (Gödel's second theorem). A human mind cannot either for itself. Nothing proves, then, that a human sees the truth of the Gödel formula: he may be a machine, perhaps an inconsistent one.

a defender replies

see the argument

John Lucas defender of the argument · 1961

Lucas answers that humans, though they sometimes err, are not inconsistent: they are fallible, and they reject an idea that leads to a contradiction. A truly inconsistent system would accept anything. He concludes that the mind can be assumed consistent.

Paul Benacerraf 1967

Benacerraf answers that Lucas's reasoning proves at most a disjunction: either the mind is not a machine, or we cannot know the machine that describes it, nor prove it consistent. If the mind is a machine, nothing requires us to be able to see the formula that escapes it.

See the rest of the debate (1)

Roger Penrose 1994

Penrose answers in Shadows of the Mind that a mathematician who believes his reasoning is sound cannot at the same time believe that a given program describes it: he would have to believe it as sound as himself, which leads to a contradiction.

Where the debate stands (our summary)The point of disagreement: do we know that we are consistent? Lucas says yes, at least in the main. His critics say human knowledge is fallible, and that this suffices to block the argument. Most logicians consider this objection decisive.

Alan Turing 1950 · attacks the conclusion

Each machine is unable to answer some questions, but each human also errs. Our superiority is felt only over the one machine we have beaten, never over all machines at once.

a defender replies

see the argument

John Lucas defender of the argument · 1961

Lucas answers that the question is not who is superior, but whether the mind and the machine are the same thing. The mechanist must offer a single machine able to do everything the mind does. For any given machine, the mind knows something it cannot.

Where the debate stands (our summary)The point of disagreement: is it enough for a human to beat each machine one after the other for him not to be a machine? Lucas says yes, since the mechanist must offer a single machine. His critics answer that the human could be a machine he cannot identify. Open point.

Solomon Feferman 1995 · attacks premise 2

Penrose assumes that a mathematician can see that the rules he uses are correct, and then derive the Gödel formula of those rules. Feferman examines this step closely: knowing that a system is correct is not something done with that system, and the exact form of Penrose's argument does not justify it.

a defender replies

Roger Penrose defender of the argument · 1996

Penrose answered his critics in an article in the journal Psyche. He maintains there that mathematical understanding cannot be replaced by an algorithm that we would know to be sound, and he spells out the steps of his reasoning.

Where the debate stands (our summary)The point of disagreement: does understanding mathematics require knowing that our rules are correct, beyond any formal proof? Penrose says yes, his critics say that is the step that would need justifying. Open point: no widely accepted answer.