Lucas’s argument is sometimes put this way: It is claimed that any machine that can produce as true only things that we can prove cannot produce as true all the things that we can prove. For everything we can genuinely prove is true, so if the machine can produce only things we can prove, everything it can produce as true is true. And all truths are consistent with each other, so the things the machine can produce as true are consistent. And we have just proved this. But the machine itself cannot prove that the things it can produce as true are consistent, by Gödel’s theorem. What (if anything) is wrong with his argument in this form?
Bridging the Gap and Burning the Bridge: The Fault Lines in J.R. Lucas’s Argumentation
What makes a mind, and what makes a machine? And can a machine ever make a mind? These are clearly important questions to ask in light of modern computing (as recently as two weeks ago, NVIDIA announced a partnership with General Motors to give all cars a mind via self-driving capabilities. Although, in the spirit of this essay, whether one needs a mind at all to drive could perhaps be easily disproved by visiting the roads of Florida, wherein there is not a mind in sight and yet they still drive, albeit poorly), but these questions were just as important in the days of Alan Turing, Kurt Gödel, and J.R. Lucas. Alan Turing first posed his imitation game in 1950, Kurt Gödel revealed his Incompleteness Theorems in 1931, and J.R. Lucas weaponized these theorems in 1961 when he published “Minds, Machines and Gödel.” Lucas argued that for any given machine, the human mind could always “see” the truth of a statement that the machine could not prove, which would thus demonstrate an unbridgeable gap between mind and machine. Our prompt offers a variation on this theme, which switches things up from the truths lying in specific Gödel sentences to the arguably more fundamental notion of consistency, leveraging Gödel’s Second Theorem. This argument tries to forge an asymmetry– we reflect upon a machine that is allegedly mirroring our proofs and declare this machine consistent. However, the machine itself is shackled by Gödel’s Second Theorem and remains incapable of asserting its own consistency. This suggests that the difference in capability proves that the mind transcends the machine. It’s a clever twist, but when held against the precise nature of Gödel’s proofs (as detailed by Nagel and Newman) and subjected to scrutiny like in lecture, this argument (like its Lucasian predecessor) reveals itself to be built upon less than stable foundations. Its conclusion rests on shaky assumptions about what is a human “proof,” the certainty of our own logical consistency, and whether Gödel’s theorems can even be readily applied in this context.
Before we start dissecting the argument, I think it’s important to navigate the space that Gödel shaped. His work was largely a response to Hilbert’s program, which was an extremely ambitious project that sought to ground mathematics in secure, unshakeable foundations. This was to be done by formalizing mathematical theories into axiomatic systems (calculi) and then proving their consistency using basic “finitistic” meta-mathematical methods. Reasoning about the system was thought to be totally transparent and utterly uncontroversial (Nagel & Newman, pp. 26-33; March 4th Lecture). Hilbert dreamt of an “absolute” proof that didn’t just shift the question of consistency elsewhere, like proving the consistency of non-Euclidean geometry by modeling it within Euclidean geometry (Nagel & Newman, p. 18).
Gödel’s masterstroke was arithmetization. He demonstrated how to assign unique Gödel numbers to every symbol, formula, and proof within a formal system that was rich enough to express arithmetic. This allowed him to then map meta-mathematical statements about the system onto arithmetical models within the system (Nagel & Newman, pp. 68-79). The statement “Formula F is provable” could be mirrored by a complex but definite statement about the properties of certain numbers..
This mapping lead to his two incompleteness theorems:
First Theorem: In any consistent formal system S strong enough for arithmetic, there’s a “Gödel sentence” G, representing “G is not provable in S,” which is true but formally undecidable within S (Nagel & Newman, pp. 6, 85-86). Any sufficiently strong, consistent system is necessarily incomplete – unable to capture all the truths expressible in its own language.
Second Theorem: Such a system S, if consistent, cannot prove the arithmetical formula A which represents the meta-mathematical statement “S is consistent” (Nagel & Newman, pp. 58, 95-97). In essence, a system cannot formally vouch for its own consistency using only its internal machinery. This result dealt a severe blow to Hilbert’s original program.
The argument expressed in the prompt attempts to exploit the above Second Theorem. Let M represent the machine’s formal system, and then let H represent the system of human mathematical proof. The steps are rather clear:
M outputs things that only H can prove.
H proves only truths, so the consistency of H holds.
Therefore M outputs only truths, thereby the consistency of M holds.
We, as H, just performed steps 1-3 which establishes the consistency of M.
However, by the Second Theorem M cannot prove the consistency of M, which shows that H has established something that M cannot.
Therefore H ≠ M.
Unlike Lucas, however, who focused on the mind “seeing” the truth of G, this argument hinges on the mind “proving” the consistency of M. This consistency-based argument, while structurally neat (and easy to follow), hits several snags.
One large problem we collide into is in step four: what is the human “proof” of the consistency of M? “And we have just proved this” Okay sure, but what kind of proof is it? If this is meant to be a formal proof within H, this argument immediately runs into trouble with Gödel. The derivation (steps 1-3) relies fundamentally on the assumption of the consistency of H. But if H is a system that the Second Theorem covers, the consistency of H is literally exactly what H cannot formally prove within itself. Therefore, any proof that relies on the consistency of H can never be a complete, formal demonstration that originates solely within H. It’s like trying to guarantee that your fortress is totally safe by using blueprints you drew while being completely inside of it, when you never went outside to check its foundations.
Well, alright, but what if this is an informal, meta-mathematical insight and not a formal proof? We’re reasoning about the systems: if H is consistent, then M, as a subsystem of H, must also be consistent. This mirrors how we understand Gödel’s own argumentation (Nagel & Newman, p. 93). But why exactly are we assuming that this kind of meta-level reasoning is beyond any possible machine? As discussed during the March 18th lecture, appeals to a special human capacity for “absolute provability” or immediate “seeing” are philosophically problematic and risk begging the question against mechanism. The argument, phrased as in the prompt, doesn’t establish why that specific insight is computationally inaccessible.
We hit another giant problem on step 2: can we even confidently assert our own consistency? The entire chain that leads to the consistency of M depends on the step where we say that “everything we can genuinely prove is true.” Is this really secure enough to ground the argument?
As discussed in the March 18th lecture, the Paradox of the Preface questions whether or not we can be utterly certain that the totality of human mathematical knowledge is perfectly free of contradiction. While we operate under the assumption of consistency, absolute certainty is elusive.
Gödel’s Second Theorem itself suggests that there is a limit to provable certainty about our own consistency, assuming that our capacity for reasoning is sufficiently strong. If we cannot formally prove the consistency of H from within H, then asserting it as an unquestionable premise in the second step seems to me overly bold for a deductive argument that aims for logical certainty. Our confidence in M’s consistency is then merely conditional: the consistency of M holds if the consistency of H holds. Is this conditional conclusion really something impossible for a machine to derive?
Finally, what machine are we actually talking about? Gödel’s theorems apply to very specific, formally defined systems. The prompt refers to “any machine,” which seems to be a problematic generalization. To rigorously apply the Second Theorem and declare that M cannot prove the consistency of M, M must be a precisely defined formal system, consistent, and sufficiently strong enough to express arithmetic. The argument casually assumes that “any machine that can produce as true only things that we can prove" automatically meets this criteria, but is this warranted? Real world computers, to say nothing of any hypothetical future technology, might operate on principles that aren’t perfectly encapsulated by the static, formalized systems that Gödel studied. Even now, we have large language models that involve processes that involve probabilistic reasoning and other learning mechanisms that don’t fit neatly into the ZFC-like framework assumed. Applying Gödel’s theorem in this way to such a vaguely defined “machine” seems unjustifiable.
Furthermore, this argument hinges specifically on how the consistency of M is proven by H versus how M might attempt it. H’s “proof” in steps 1-3 is meta-mathematical and relies on assessing the relationship between H and M and the assumed nature of H’s proofs. M, when trying to prove its consistency, would presumably need to do this formally using its internal axioms and rules– something that Gödel prohibits. The argument thus compares an informal, external assessment by H with a failed formal, internal attempt by M. I do not think that this comparison establishes a fundamental difference in capability, but rather something along the lines of a difference in perspective along with the type of reasoning that is being employed. It doesn’t seem to me clear that a sufficiently advanced machine couldn’t also perform meta-level analyses of its subsystems or even itself— arriving at a conditional consistency proof (e.g., if my underlying logic is sound, then system M is consistent), which isn’t ruled out by the argument.
In conclusion, while the consistency-based argument presented by the prompt offers an interesting variation on Lucas’ original theme, it also inherits similar vulnerabilities. It rests upon ambiguous notions of human “proofs” of consistency, which could run afoul of Gödel’s Second Theorem if we interpret it formally, or begging the question if we interpreted it as a uniquely human informal insight. It relies on an assertion of human consistency which is philosophically debatable and arguably formally unprovable while operating within our own system of reasoning. Finally, it applies Gödel’s highly specific theorem to some vaguely defined machine without any justification that the machine meets the necessary criteria for Gödel’s theorem. The attempt to create some unbridgeable gap between mind and machine based on proving consistency and leveraging Gödel’s Second Theorem ultimately stands on incomplete foundations that prevent it from being a conclusive demonstration.
Bibliography
Nagel, Ernest, and James R. Newman. Gödel’s Proof. New York University Press, 1958.
Lucas, J. R. “Minds, Machines and Gödel.” Philosophy, vol. 36, no. 137, 1961, pp. 112-27.