Rationality Requires Other Minds
A psychiatrist's reading of Gödel.
Here is a sentence:
Andrew Smith cannot judge this statement to be true. I (me) cannot judge this to be true, but YOU (reader) can.
The sentence is from a notebook of mine, written several years before I had any thought of publishing on it. It is a Smullyan-style construction that I owe, by way of its credit in the margin of that page, to David Deutsch.
Try the puzzle from inside my head. If I judge the sentence true, I have judged that I cannot judge it true. If I judge it false, I have judged that I can judge it true, which I have just demonstrated that I cannot do. From where I stand, the sentence is a wall.
From where you stand, reading it, the sentence is plainly true. You can see what I cannot.[1]
An echo from arithmetic
Gödel’s first incompleteness theorem, in plain terms, is the result that any formal system rich enough to do arithmetic contains true statements that cannot be proven from within. The sentence above is a Smullyan-style cousin of the kind of self-referential proposition Gödel built. There is a tidy genealogy for the form, running from C. H. Whiteley’s original construction against Lucas through Smullyan’s Forever Undecided (1987) and Deutsch’s The Beginning of Infinity (2011), and from there into my notebook.[2] I take no credit for inventing it.
How to apply the formal result of Gödel’s incompleteness theorem to matters of the mind, if it applies at all, is where academic readings begin to diverge.
Roger Penrose reads Gödel as evidence that human mathematical insight cannot be entirely algorithmic, and so cannot be what a Turing machine does.[3] Douglas Hofstadter, in Gödel, Escher, Bach and I Am a Strange Loop, reads Gödel as a model of how an “I” emerges from a substrate at all, by way of self-referential loops within a single brain (the eponymous “strange loops”).[4] The two readings are different from each other, but they share an important feature: each of them stays inside one mind. Penrose gives that mind a non-algorithmic faculty; Hofstadter gives it a recursive one. In each case, what Gödel means in the context of the mind is about a single mind, working alone.
As a psychiatrist, I read Gödel in a third way. The same incompleteness that makes the inner “I” possible for Hofstadter also makes that “I” insufficient for certain judgments about itself. The mind that wants to judge a Gödel-style proposition about itself runs into the wall I described above, and the resolution does not arrive by way of a more powerful version of the same mind. It arrives by way of another mind, standing somewhere the first mind cannot, doing the judging from there.
Several acknowledgments are necessary before I go further. David Deutsch is the closest predecessor I know of for this argument. His “tradition of criticism” depends on other people for error correction, with fallibilism as the underlying motivation, and he treats Gödel as a special case of computational limits. He almost reaches the argument I am making here, but stops just short of it.[5] In a 1998 paper in the Bulletin of Symbolic Logic, Stewart Shapiro made a related point at the level of mathematical practice itself: no individual mathematician produces the full Gödel-extension; the community does. Shapiro’s argument is a logical-defect critique of the Lucas-Penrose program; my argument goes slightly further, treating the social as constitutive of rationality rather than merely corrective of it.[6]
Solomon Feferman’s “Penrose’s Gödelian Argument” (1996) is the canonical mathematician’s rebuttal to Penrose, arguing that mathematicians proceed by trial and idealization rather than by surveying any formal system whole. Hilary Putnam, going back to “Minds and Machines” (1960), pressed a different objection: the Lucas-Penrose argument applies, if anywhere, to mathematics as a whole rather than to individual human reasoners (who are inconsistent). Peter Koellner’s two-part 2018 Journal of Philosophy paper is the current state-of-the-art formal treatment, and finds Penrose’s later argument invalid. The argument I am making here sits to the side of all three of these critiques. They are objections to the Lucas-Penrose claim that the mind exceeds any machine; I am arguing that no mind, exceeding any machine or not, can complete its own Gödelian self-judgments alone.[7]
On the clinical side, recent philosophical work on delusion, including Miyazono and Salice’s 2021 paper in Synthese, treats delusion within a social epistemology rather than as a private cognitive failure. Richard Klein’s 2009 reading of Lacan-on-Gödel in The Symptom sits in a similar neighbourhood.[8]
About the analogy
The argument from formal systems to minds is an analogy. The strict version of Gödel’s result requires a formal system meeting specific conditions, and nothing in psychiatry or in everyday life meets those conditions in the strict sense. I am not asserting that a person is a Turing machine, or that the human mind is actually a formal system in Gödel’s terms. My claim is this: any reasoner whose access to its own premises is partial (which is every reasoner, formal or fleshly) will have blind spots that are visible from elsewhere and invisible from inside. The Gödel result is a clean limit case of a more general phenomenon, formalized inside a system tidy enough to admit mathematical proof. Outside that system, what we have is a less tidy version of the same situation. Whether it is rigorously the same situation is exactly the place a careful reader can object, and they would be right to do so.
The thesis sentence I am working toward is the strong one even so. From the same notebook:
No system, not even my conscious mind, can be perfected such that its internal logic cannot be thwarted by an externally true but unprovable statement, or rather, especially not. Rationality requires other minds.
The contention buried within “or rather, especially not” is that the case where the mind is the system in question is the case that Gödel’s incompleteness is most relevant for, not least. To be rational, a reasoner must do more than maintain internal consistency. They must be able to recognize the limits of their own reasoning. Recognizing those limits may feel achievable in the day-to-day, superficial sense, but at any meaningful depth, parsing out the limits of one’s own reasoning using only one’s own reasoning to do so may begin to remind you of the wall of logic I ran into at the beginning of this piece.
A few clarifications
I am not claiming that truth is socially constructed. The relevant feature of the other mind in this argument is positional: it sees from somewhere I cannot. Whether it agrees with me is incidental, and a consensus of minds in identical positions to mine would not help. What matters has to do with where the second mind is positioned, not with how many minds happen to share its conclusion.
I am not claiming Hofstadter’s strange-loop story is wrong. Hofstadter uses Gödel to explain how the inner “I” emerges from a substrate at all. I am using Gödel to explain why that “I”, however it emerged, is insufficient for certain judgments about itself. These are fundamentally different questions, and Gödel’s work may well be relevant in both contexts.
There is a passage in Hofstadter’s 1999 preface to Gödel, Escher, Bach where he raises the prospect of formal systems modeling other formal systems but then notes that any sufficiently powerful system already contains models of infinitely many other systems. These internal models (even infinitely many of them, all the way down)[9] cannot validate their own self-referential propositions; an external mind is still needed, no matter how many recursions we nest in there.
Elsewhere in the same book, in the “Gödel’s Theorem and Personal Nonexistence” section of Chapter 20, Hofstadter denies that one can import an outsider’s view of oneself into oneself.[10] He is right about this. The “outsider’s view” he forecloses is one you would have to inhabit, by mapping yourself onto another person from inside your own mind. The argument I am making does not require that; it requires receiving a judgment another mind has made from its own position.
Reordering
Most of us learn somewhere along the way that reasoning is something you do alone and then bring to others for checking. The order is: think first, talk second. This is workable. For a sizeable class of propositions it works perfectly well, and I am not arguing against it for those propositions.
What I am arguing is that for another class of propositions, mostly the ones about ourselves, the order runs better the other way around. The other mind comes earlier. For those propositions the work is distributed across more than one mind from the start, because the propositions in question are exactly the kind a closed system cannot settle from inside. In those cases, the other mind is where the reasoning occurs; conversation is how it gets received.
Two friends arguing heatedly late at night about something one of them cannot quite see about themselves. A clinician and a patient sitting with a question the patient cannot answer from where they sit. A musician in the studio so deeply focused on the nuances of a recording they're working on that they can no longer tell whether it is any good. These three scenes are instances of the same epistemic situation. In each one, a second mind is needed to do something the first one cannot do alone. The reason has nothing to do with effort. It has to do with where each mind is standing.
Rationality requires other minds.
Notes
1. The closest non-Gödelian precedent for the I/YOU asymmetry in the cold open is Wittgenstein’s private language argument: Ludwig Wittgenstein, Philosophical Investigations, trans. G. E. M. Anscombe (Blackwell, 1953), §258 and surrounding remarks. I am bracketing the separate dispute over Wittgenstein’s Remarks on the Foundations of Mathematics on Gödel itself, partially rehabilitated by Juliet Floyd and Hilary Putnam, “A Note on Wittgenstein’s ‘Notorious Paragraph’ about the Gödel Theorem,” Journal of Philosophy 97, no. 11 (2000): 624-632.
2. The puzzle’s sentence form descends from C. H. Whiteley, “Minds, Machines and Gödel: A Reply to Mr Lucas,” Philosophy 37, no. 139 (1962), where “Lucas cannot consistently assert this sentence” was deployed as a tu quoque against Lucas. Raymond M. Smullyan, Forever Undecided: A Puzzle Guide to Gödel (Knopf, 1987), built the form into a book-length treatment of Gödelian self-referential puzzles. David Deutsch, The Beginning of Infinity: Explanations That Transform the World (Viking, 2011), reused it in his discussion of computational limits.
3. Roger Penrose makes the Gödelian argument most fully in two volumes: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford University Press, 1989), and Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford University Press, 1994). The second develops the argument that human mathematical insight cannot be entirely algorithmic more rigorously than the first, and is the volume the formal rebuttals (see Note 7) primarily address.
4. Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, 1979; 20th-anniversary edition with new preface, 1999), is the foundational book-length treatment of how an “I” emerges from a substrate via self-referential strange loops within a single brain. The argument is restated and extended in I Am a Strange Loop (Basic Books, 2007).
5. David Deutsch, The Beginning of Infinity: Explanations That Transform the World (Viking, 2011), develops the “tradition of criticism” as an epistemological framework grounded in fallibilism, with Gödel’s incompleteness treated as a special case of computational limits and error correction depending on other people.
6. Stewart Shapiro, “Incompleteness, Mechanism, and Optimism,” Bulletin of Symbolic Logic 4, no. 3 (1998): 273-302.
7. Solomon Feferman, “Penrose’s Gödelian Argument: A Review of Shadows of the Mind by Roger Penrose,” Psyche 2 (1996): 21-32, is the canonical mathematician’s rebuttal to Penrose, arguing that mathematicians proceed by trial and idealization rather than by surveying any formal system whole. Hilary Putnam’s earlier and broader objection appears in “Minds and Machines,” in Sidney Hook, ed., Dimensions of Mind: A Symposium (New York University Press, 1960), 138-164. Peter Koellner, “On the Question of Whether the Mind Can Be Mechanized, I: From Gödel to Penrose,” Journal of Philosophy 115, no. 7 (2018): 337-360, and “II: Penrose’s New Argument,” Journal of Philosophy 115, no. 9 (2018): 453-484, give the current state-of-the-art formal treatment.
8. Kengo Miyazono and Alessandro Salice, “Social Epistemological Conception of Delusion,” Synthese 199 (2021): 1831-1851. Also: Richard Klein, “Lacan and Gödel,” The Symptom 10 (Spring 2009); Klein is a Lacanian Ink contributor rather than an academic philosopher, and the link-through reflects that.
9. The phrase “all the way down” is a deliberate echo of “turtles all the way down,” an idiom for infinite regress that descends from an apocryphal cosmological anecdote about a series of turtles supporting one another indefinitely. The attribution is contested; William James and Bertrand Russell are the names most frequently invoked, and Stephen Hawking opens A Brief History of Time (Bantam, 1988) with one familiar version. Internal modeling can be iterated indefinitely without ever exiting the system, but no depth of internal modeling supplies the external standpoint the argument requires.
10. Douglas R. Hofstadter, Gödel, Escher, Bach (Basic Books, 1979), Chapter 20: “Strange Loops, or Tangled Hierarchies.” The “Gödel’s Theorem and Personal Nonexistence” section begins on p. 698 of the original edition.