Lucas-Penrose Argument

Date: January 16, 2026

There is a popular argument against mechanism that uses Gödel’s Incompleteness Theorem, endorsed by Lucas and Penrose.

The argument runs as follows: define mechanism as the claim that the human mind is a computational process. For the sake of argument, let us suppose that mechanism is equivalent to the claim that the mind is a Turing machine. Suppose that the mind is a Turing machine $M^{\prime }$. Then there is some theorem-proving Turing machine $M$ such that $M$ is equivalent to $M^{\prime }$, so we may replace $M^{\prime }$ by $M$. Now, there is some formal system $F$ whose theorems are those and only those statements that $M$ proves. Assuming that $F$ is consistent and applying Gödel’s Incompleteness Theorem, there exists a sentence $G$ of $F$ such that neither $G$ nor $\neg G$ are provable in $F$. This means that there exists a sentence $G$ of $F$ such that $M$ cannot prove $G$ and $M$ cannot prove $\neg G$.

Now, as the argument goes, since the human mind can know that $G$ is true (in the standard model of arithmetic), but $M$ cannot prove $G$, the human mind can recognize mathematical truths that $M$ cannot. Since $M$ was arbitrary, the human mind can recognize mathematical truths that no Turing machine can. Therefore, the human mind cannot be a Turing machine.

To start, the argument seems to conflate truth and provability. Namely, it states that the human mind can know that $G$ is true, but $M$ cannot prove $G$. This is true, but the argument must show that the human mind can prove $G$ in order to conclude that the mind is not a Turing machine. It has not shown this.

Another issue with the argument is the unjustified assumption that $F$ is consistent. It might be the case that $F$ is inconsistent.

References

• Paul Benaceraff, God, the Devil, and Gödel. https://eclass.uoa.gr/modules/document/file.php/PHILOSOPHY1048/Benaceraff.%20God%2C%20Devil%2C%20Godel.pdf

Kurt Gödel Paul Benaceraff Roger Penrose John Lucas Gödel’s Incompleteness Theorem