Gödel’s incompleteness theorems shook the foundations of mathematics, revealing its inherent limitations. These discoveries are now among the most significant in the field and have inspired various philosophical viewpoints. One influential perspective, proposed by Nobel Prize-winning physicist Sir Roger Penrose, is that Gödel’s theorems prove the mind is not a computer. However, LSE philosopher and logician Wesley Wrigley argues against Penrose, asserting that Gödel’s theorems do not imply the mind is non-computational. Join Wes as he demystifies the Penrose argument and explains why, despite Penrose's brilliance, it is a mistake to claim on the basis of the incompleteness theorems that the mind cannot be implemented by a computer.

Gödel’s incompleteness theorems are among the most important discoveries in mathematics. Unlike most discoveries in mathematics, they have been ascribed a dizzying array of profound implications by mathematicians, scientists, and philosophers. One alleged philosophical implication is that the human mind, in its mathematical capacities, must be non-computational. If this “anti-mechanist” view is correct, then no computer algorithm can fully simulate the mathematical reasoning that human beings are capable of. In addition to telling us something significant about ourselves, this would seem to refute the possibility of a true artificial intelligence, conceived of as a computer that has the same cognitive powers as a human being. The most persuasive argument for anti-mechanism on the basis of Gödel’s theorems is due to distinguished physicist Sir Roger Penrose. My aim is to give you a clear idea of what his argument is, and why I think it is unsuccessful.

The First Incompleteness Theorem

A system of arithmetic comes in two parts. The first is a precisely specified list of initial formulae, the axioms. Usually, axioms are obvious truths that we take ourselves to need no proof of, such as ‘no natural number comes before 0’. The second part is the class of theorems, formulae that can be derived from the axioms by repeatedly applying the rules of formal logic. A system is sound if all the axioms, and hence the derived theorems, are true. For Penrose’s argument, we need only a basic version of Gödel’s first incompleteness theorem: no sound system of arithmetic can be used to derive all the true arithmetical formulae. Each such system is fundamentally incomplete.

Indeed, for a suitable system S, we can find a particular arithmetical truth that cannot be derived in S, namely S’s Gödel sentence. This formula, Gs, essentially says (via coding) that Gs is not derivable in S. So, if Gs is derivable in S, it is false. Hence, if S is sound, Gs is not derivable in S. But that means Gs is true, so S is incomplete. The situation is unavoidable: although we could adopt Gs as a new axiom, the expanded system would have its own Gödel sentence which would be true, but unprovable in that system.

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There is an insurmountable limitation on the mathematical powers of any computer. If human beings do not share this limitation, then our mathematical abilities are essentially non-computational, no matter how superior a computer might be in terms of speed, accuracy, or memory.

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 What is the connection between Gödel’s theorem and a computational account of the mind? Let C be a computer that simulates the process of deriving arithmetical formulae. Roughly speaking, C will repeatedly apply logical operations to some precisely specified input formulae, according to an algorithm. This means that the total output of C (if left to run indefinitely) will exactly match some system of arithmetic, Sc. Anthropomorphising somewhat, we say that C can derive any arithmetical statement which is a theorem of Sc, and vice-versa. Gödel’s theorem then tells us that if C derives only true formulae of arithmetic, it cannot derive its own Gödel sentence (i.e. Gsc), which is therefore true. So, there is an insurmountable limitation on the mathematical powers of any computer. If human beings do not share this limitation, then our mathematical abilities are essentially non-computational, no matter how superior a computer might be in terms of speed, accuracy, or memory.

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Penrose’s Argument

Penrose’s best anti-mechanist argument is the “central new argument” in his Shadows of the Mind. It is quite subtle, so we shall work through it with some care.

First, suppose that all the methods of reasoning which I can use to produce (what Penrose calls) unassailable arithmetical proofs are simulated exactly by a theorem-deriving computer M. (This supposition will, according to Penrose, lead to a contradiction, and hence must be wrong.) We know that the output of M must exactly match some system of arithmetic. But which one? Let F be any such system, and consider the hypothesis that a formula is provable by me if, and only if, it is a theorem of F. Assuming that I am simulated by M, this can be given a rigorous formulation, abbreviated as I = F.

Under the hypothesis that I = F, every theorem of F is something which I can unassailably prove. Since what I can unassailably prove must be true, it follows that F is sound. Now consider the system F+, which is like F, but with I = F as an additional axiom. Since F is sound and we are still under the hypothesis that I = F, it follows that F+ is sound as well. By Gödel’s theorem, Gf+ must be true.

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All the above reasoning was under the hypothesis that I = F. We can thus conclude that if I = F, then Gf+ is true. This is something I know; indeed, I have just given a proof of it. This is where the trouble starts. For suppose again that I = F. This means that F and F+ are sound, and that everything I can prove, including the statement “if I = F, then Gf+ is true”, is derivable in F. Since F+ is a more powerful system, that statement is also derivable in F+. But in F+, I = F is an axiom. Hence Gf+ is derivable in F+ itself. That contradicts Gödel’s theorem, since F+ is a sound system.

This contradictory situation arose because we assumed that I = F; hence that assumption must be rejected. But F was an arbitrarily chosen arithmetical system. So, the class of formulae that I can prove does not exactly match the theorems of any arithmetical system. This in turn contradicts the supposition that I am exactly simulated by M, since the output of M must exactly match some arithmetical system. Since M is an arbitrary theorem-deriving computer, it follows that no such computer simulates me exactly.

Penrose’s Argument Reconsidered

The argument is ingenious. If it works, it tells us something profound about our own minds, and the limitations of any possible computer simulation of them. But the argument seems to me to have a fatal flaw. During the argument, we need to establish that I can prove that if I = F, then Gf+ is true. During my proof, I need to demonstrate that F is sound (under the assumption that I = F). The key premise at this point in my proof is that anything I can unassailably prove must be true. Call this the truth premise. It is the critical weak spot in the argument.

We must be very careful about the role played by the truth premise in Penrose’s argument. It is not directly a premise of the argument at all. Rather, it is a premise in my proof that if I = F, then Gf+ is true. So Penrose’s assumption is actually that the truth premise is the sort of thing that one can use in an unassailable mathematical proof. Since the conclusion of such a proof must be unassailable, the premises must be too, otherwise there would be room for some degree of rational doubt about the conclusion. But is the truth premise really unassailable, in the same sense that basic arithmetic is usually taken to be?

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It also seems that numbers cannot be mental entities either: there are infinitely many primes, but how could a finite being like me have infinitely many ideas in their mind? So, it looks as though the truth premise implies that there are infinitely many peculiar entities existing outside of space, time, and thought.

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Penrose argues that the truth premise is highly plausible, and that in using mathematics to prove new theorems which we really believe, we commit ourselves to accepting it. I certainly agree that it is plausible. I am also sympathetic to the commitment idea; if we did not think that everything we can unassailably prove is true, it would seem irrational for us to consider unassailable proof to be such a compelling reason for belief. But neither plausibility nor intellectual commitment ensure the required unassailability of the truth premise.

The truth premise is not an ordinary mathematical belief, like 1 + 1 = 2, that we might describe as unassailable. It does not concern numbers, function, or graphs, but rather the relation of our mathematical beliefs to reality. As such, it has rather significant philosophical consequences. One of the most striking is the existence of mathematical objects. For example, I can give an unimpeachable proof of the elementary theorem that there are infinitely many prime numbers. If the theorem is true, we must consider what these numbers are. They cannot, it seems, be physical objects, since it makes no sense to ascribe physical properties like mass, charge, or position to them. It also seems that numbers cannot be mental entities either: there are infinitely many primes, but how could a finite being like me have infinitely many ideas in their mind? So, it looks as though the truth premise implies that there are infinitely many peculiar entities existing outside of space, time, and thought.

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Penrose’s argument is almost convincing. But it fails to respect a crucial distinction between the impressively compelling justification that we have for ordinary mathematical beliefs, and the weaker justification that we have for philosophical beliefs about those mathematical beliefs.

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Consequently, many philosophers argue that mathematics is not true after all. They might be wrong, but the fact that they have a serious case means that the truth premise is not unassailable. So my attempt to unassailably prove that if I = F, then Gf+ is true, is a failure. Hence, I cannot infer from the assumption of I = F that Gf+ is derivable in F+. Since that is what would contradict Gödel’s theorem, it does not follow that it is impossible that M simulates my mathematical abilities.

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Penrose’s argument is almost convincing. But it fails to respect a crucial distinction between the impressively compelling justification that we have for ordinary mathematical beliefs, and the weaker justification that we have for philosophical beliefs about those mathematical beliefs. Beliefs of the latter kind, like the truth premise, are not unassailable, even if they can be convincingly argued for. Several great thinkers, including Gödel, have hoped that philosophical questions might ultimately be answerable with mathematical certitude. To date, philosophy has not lived up to this standard.