The Limits of Self-Knowledge
What can a sensible treatment of Gödel teach us about self-knowledge?
Introduction
Descartes’ epistemological project is one which aims to find a safe point immune from radical doubt. The first refuge Descartes finds is in the cogito, which is “necessarily true whenever it is put forward by me or conceived in my mind” (Descartes, 1641/1984, p.17). Descartes extends this point to arrive at the rule that “whatever I perceive very clearly and distinctly is true” (Descartes, 1641/1984, p. 24). In some way, according to Descartes, the mind has some epistemic superpowers. The external world is misleading and thus rational to doubt, yet the thinking thing has the superpower of knowing that it is available completely to itself.
Understandably, many doubt Descartes’ descriptions of the abilities of the mind. More modest Cartesians say only some mental acts are known with a certainty not available in any empirical knowledge. For example, I might see the coffee mug in front of me, but I can’t see the coffee mug in its entirety. I’m missing its backside, its bottom, and its interior. My mental act of perceiving the mug, however, is something I know I am having and can “see from all sides,” so to speak. The stronger Cartesian thesis, which I will argue against and call strong Cartesian self-transparency, says that the mind can fully survey and certify itself through reflection. This means the mind has complete and utter privileged access to its own mental states, complete transparency in understanding its own mental states, and crucially, the right to determine how the mind knows itself, and what I will call reflective completeness. A mind being reflectively complete entails that in reflection it has the cognitive resources necessary to understand and answer adequately all questions about its own mental states, its epistemic authority over its own cognitive practices, and its standards for self-knowledge.
I will argue against this stronger claim by gesturing to Gödel’s Second Incompleteness Theorem. Let me make clear that the following arguments do not take Gödel’s Second Incompleteness Theorem to be a result that concerns philosophy of mind. Instead, Gödel’s theorem, which is the term I will use to refer to Gödel’s Second Incompleteness Theorem henceforth for the sake of space unless otherwise noted, has to do with formal systems which have enough expressive strength to represent basic arithmetic (Gödel 1931/1986). However, what I suggest Gödel’s Theorem gives a philosopher interested in self-knowledge is a structural moral about self-reference and the process of internally certifying our beliefs.
In this essay, I will argue that when we restrict the epistemological significance of Gödel’s Theorem to an analogical one, we can derive what I will call the Gödelian Self-Knowledge Moral, which says that when a sufficiently complex thing can represent itself, and when it tries to certify its own adequacy using only first-personal resources, it will face a structural limit. I will apply this moral to the stronger Cartesian understanding of self-knowledge. I argue that even if we take the Gödelian moral seriously, it is still too weak to make a dent in first-personal privilege broadly speaking, but undermines “reflective completeness,” as well as the ideal of exhaustive self-transparency.
The deployment of the analogical argument, and the use of Descartes in his strongest form, is intentional. Many interpretations of Gödel’s proof are outlandish and take Gödel to be making a philosophical claim,(1) when in fact, he is making a purely formal point (Franzén 2005). However, Gödel’s use of “self-reflection” is fascinating, and has interesting parallels to several areas of philosophy, music, and literature, as Hofstadter has noticed (Hofstadter, 1979, 2007). One of the aspects which makes Gödel incredible is his formalization and arithmetization of this concept of self-reflection and its propensity toward producing incompleteness. Drawing an analogy from the structure of Gödel’s formalization seems like it could at the very least be interesting, if not useful, and we ought to test it out while remaining cognizant that the proof, as Goldfarb says, “is purely number-theoretic and uses only simple arithmetical principles” (Goldfarb, 2025, p. 48). More on that to follow.
Returning to the point, if we are to begin someplace with using Gödel’s proof in philosophical contexts, it must be done in the weakest form possible, in our case, analogical argument. After all, one will not win favor with an analytic philosopher or logician if they are to make wildly strong arguments in epistemology or philosophy of mind using Gödel that simply do not follow from his work. Moreover, taking Descartes at his strongest on self-knowledge, a notoriously easy-to-object-to view, will allow us to see if even the weakest philosophical interpretation of Gödel can make a dent in a view meant to be deflated. If the project fails, we ought to let Gödel’s great proofs lie in the realm of pure logic. If the project is successful, then it proves to be a sensible starting point for any philosophical treatment of Gödel’s work.
Analogical Argument
Given that Gödel’s proof does not warrant a direct inference from formal incompleteness to the opacity of the mind, we will need to find another route to make our point.
One way to get around this problem is through analogical reasoning. Bartha characterizes this as an ampliative process rather than the typical deductive reasoning we employ in philosophy. Conclusions are supported by relevant similarity, rather than strict logical-style deduction which guarantees validity (Bartha, 2010). For the sake of clarity, I will make clear that there are three ways in which my analogical argument should not be understood. First, it does not claim directly that Gödel refutes Cartesianism. Second, it does not claim that the mind ought to be identified as a formal system, and third, it does not claim the opposite as a result, that the mind must be stronger than a formal system, as Lucas and Penrose have argued (Lucas, 1961; Penrose, 1987).
As we learned from Gödel, formal systems that represent their own syntax and attempt to prove their own consistency will always run into a certain type of failure when doing complete internal certification. Analogously, a subject that represents its own epistemic standing and tries to certify the adequacy of its standing using its own cognitive resources faces a different, namely non-logical or mathematical, type of failure. While the problems that Peano arithmetic faces are different from that of the agent, the form of the problem is the same. Self-representation and the process of internal certification pose peculiar problems.
Putnam makes a similar sort of argument, although he does not utilize analogical reasoning explicitly in reference to interpreting Gödel’s proof (Putnam, 1994). He argues that if there were “a complete computational description of our own prescriptive competence,” then we could not come to believe that description to be correct while also being beholden to it.
Strong Cartesian Self-Transparency
Let SCST be the acronym for Strong Cartesian Self-Transparency. I define it as the conjunction of the following five conditions:
C1(Privileged Access): A subject can know at least some of its own present mental states without needing to depend on external perception.
C2(Epistemic Security): A priori first-personal knowledge is resistant to all forms of doubt.
C3(Transparency): If a subject is in a mental state, then the subject knows they are in that mental state.
C4(Internalism): Whether a piece of first-personal self-knowledge is something the subject knows about itself can be assessed reflectively from the subject itself.
C5(Reflective Completeness): There is no limit from the first-personal standpoint on what the subject can act as the truth-maker for, concerning its own present mental and epistemic standing.
C1 comes from the cogito, which is of course true a priori whenever uttered, and secured from the act of thinking itself. C2 follows from what Descartes eventually gets to by his third meditation, which is the special certainty that arises from clear reflection. C3, or our ability to be aware of the mental state that we are in, comes from Descartes’ claim that “there can be nothing in the mind, in so far as it is a thinking thing of which it is not aware,” and that “we cannot have any thought of which we are not aware at the very moment when it is in us” (Descartes, 1641/1984, p. 171). I take C4 to be evident from the methodology of the Meditations. One begins from reflection, only takes clear and distinct perceptions seriously, and certifies knowledge to that standard. This is internalism by definition. The final condition, C5, takes more explaining given that it is not a concept Descartes mentioned explicitly. I take it to be the case that C3 ∧ C4 → C5.
Read in natural language: if mental states are something we can be aware of, and if reflection using internal resources can certify knowledge of mental states, then the mind can fully grasp its own epistemic standing internally. This is meant to be the strongest interpretation of Descartes’ Meditations, and as noted, by no means do I believe that such an interpretation is correct strictly speaking. It is presented for the sake of measuring the strength of the analogical reasoning.
Gödelian Moral
Gödel’s proof says if F is a consistent formal system which has the strength to express elementary arithmetic, then F does not syntactically entail Con(F). The Gödelian Self-Knowledge Moral (GM henceforth) says if a sufficiently strong (or perhaps rich) subject represents itself and attempts to do a complete internal certification of its own epistemic standards, then some questions about that certification cannot be settled internally to that subject (or standpoint).
GM is of course weaker than Gödel’s proof given that it is not formal and, in some sense, replaces proof with epistemic certification. The formal systems are also reflective subjects, hence we get the analogy. In Gödel’s proof, the system expresses so much as to encode facts about its own proof, but when a consistency claim is formulated, it cannot prove that claim using its own resources. In the Cartesian case, the subject can formulate questions about its own epistemic status, and it can also ask whether its own resources are adequate in answering all questions about itself. If SCST is true, the answer must be in some way internally available to the subject. GM denies this possibility.
GM shouldn’t be taken as an undermining of all forms of self-knowledge given that it does not address the cogito and the cogito doesn’t certify the adequacy of its own epistemic standpoint. The condition GM problematizes is C3 and C4, and given that the falsity of either of them entails the falsity of C5, it is a problem for C5 as well.
Argument
Formally, the argument is as follows:
P1. If SCST is true, then C5 is true
P2. If C5 is true, then a subject can be the truth-maker for every question about its epistemic standing using its internal reflective powers.
P3. There exists a relevant question which concerns the adequacy of a subject’s reflective resources in certifying the adequacy of its self-knowledge. This is a Cartesian Gödel sentence, so to speak.
P4. Any subject that can answer that question using its own resources is self-representing and can undergo complete internal certification of its own epistemic standards.
P5. By GM, there does not exist a strong enough self-representing subject which can guarantee its own complete internal certification of its epistemic standards.
C. SCST is false because C5 is false.
The premise of interest is P3. If SCST were weaker, then the second-order questions about its ability to certify its epistemic status would not be a problem. One could respond by saying they know they are thinking and that they don’t need to consult the external world due to this fact. The strength of SCST makes it such that the first-person subject has a privileged epistemic security whose adequacy is knowable from within. Once that strong claim is realized, the second-order question of P3 immediately follows and is fair game.
Hence C5 is the problem condition. C1 and C2 assert the immediacy and security of self-knowledge, so they are not within GM’s crosshairs. C3’s status is unclear given that while unfolding states might be transparent to us, complete transparency seems unrealistic. C3 matters less for our purposes, so I will leave its treatment there.
In the wake of GM, we might say that a minimal Cartesian account survives, which asserts that C1∧C2∧C3 equal the minimal account. This is compatible with GM and says roughly that a subject may know what they are thinking and may know their sensations and thoughts from a first-personal perspective, and may enjoy authority in most contexts, but they lack exhaustive authority and internal certification of their perspectives and their process of perspectives.
Conclusion
The result of our stress test of the easy-to-object-to SCST using the weak analogical argument from Gödel’s proof seems to be a tentative success. There is at least some interpretation of his results which warrants some concern about self-reference, particularly in the context of self-knowledge. I believe that any future treatment of Gödel’s proof should continue on a similar track, not looking to take philosophical results directly from the formal reasoning. Perhaps I have done too much of that here, but regardless, the paradoxical nature of self-reference remains a persistent problem across philosophy and logic, and Gödel’s proof provides useful material to illuminate that fact.
Footnotes
(1) For the sake of not painting a picture of a “mechanized mind,” I will not try to make direct comparisons between functions of the mind and notions within Gödel’s construction in mathematical logic. However, here seems like an appropriate time to draw an analogy to Gödel’s Second Incompleteness. Reflective Completeness is the closest to a system “proving its own consistency.” In Gödel’s proof this is mirrored by the Con(T) function, where Con(T) expresses that the theory T is consistent. The recursivity of the Con function would mirror the survey and certify process of the mind in checking its own mental states and the internal verification practices which determine that certification process. Taken in another direction, the self-reference in the Cartesian case would be done by a sort of mental diag(n) function or substitution function Sub(n, n), which would take the Gödel number of a formula with a free variable and return a Gödel number of the sentence formed by substituting the formula’s own Gödel number into itself, thus constructing a sentence that talks about itself. Again, this is for the sake of analogy and should not be taken as a direct comparison between Gödel’s proof and the mind.
Works Cited
Bartha, P. (2010). By parallel reasoning: The construction and evaluation of analogical arguments. Oxford University Press.
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Goldfarb, W. (2025). Notes on metamathematics. Harvard University.
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Putnam, H. (1994). Reflexive reflections. In J. Conant (Ed.), Words and life (pp. 416–427). Harvard University Press. Original work published 1985.