As violins wax maudlin, viewers of the BBC documentary Dangerous Knowledge are introduced to "a small group of ... brilliant minds," who in the nineteenth- and twentieth-centuries, "unraveled our old, cozy certainties about math and the universe." These brilliant minds we are told, unraveled certainties that were so unassailable and beguiling that "once they had looked at these problems, they could not look away, and pursued the questions to the brink of insanity, and then over it to madness and suicide." Here, and through the rest of documentary, the presenter suggests a direct, and even mono-causal relation between deep mathematics and deep disturbances of mind, as apparently revealed by the life and works of four brilliant thinkers, including the mathematician and philosopher Kurt Gödel (1906 - 1978).
Gödel, it is true, exhibited paranoia, and the idea that mathematics induced his mind to madness is certainly a romantic one, suggesting the image of a tragically brilliant Narcissus, who observed the reflection of logical structures so beautiful he could not turn away, and was thereby psychically diminished. Quite apart from romanticism however, this begs the question: is this so? Can mathematics push a mind "to the brink of insanity, and then over it"? Is mathematics dangerous knowledge?
The query seems an odd one, however, the BBC presenter himself is fully solemn when making his claims, and in making such claims he is not alone. Many observers ponder the connection between deep mathematics and psychology, and, taking Gödel's work as inspiration, one biographer has suggestively entitled her book Incompleteness: The Proof and Paradox of Kurt Gödel, therein invoking Gödel's famous incompleteness theorem in connection to madness, stating;
"Gödel's theorems are darkly mirrored in the predicament of psychopathology: Just as no proof of the consistency of a formal system can be accomplished within the system itself, so, too, no validation of our rationality -- of our very sanity -- can be accomplished using our rationality itself. How can a person, operating within a system of beliefs, including beliefs about beliefs, get outside that system to determine whether it is rational? If your entire system becomes infected with madness, including the very rules by which you reason, then how can you ever reason your way out of your madness?"
Here the suggestion is not that mathematics is necessarily dangerous, but that the mathematical results of Gödel's incompleteness theorem might in some way help us unpack the predicament of "a person, operating within a system of beliefs" whose "entire system becomes infected with madness." Again, we must ask: is this so?
Quite apart from claims proposing some mathematics of madness, the question of how incompleteness bears on the mind merits consideration, and, keeping one eye on the connection between math and human states of mind, we will here examine the basic implications of Gödel's incompleteness theorem for the mind. Because we are concerned with the significance of the incompleteness theorem as it relates to what humans may know as regards the mind, we will first expound the claims and conclusions of the theorem, then discuss the implications that do (and do not) follow from the theorem, and follow this with observations on what we may reasonably believe the theorem tells us about human thought and the mind.
Here we begin with an exegetic account of the content and significance of the theorem, including a limited discussion of the mathematical details.
Collectively, the term "incompleteness theorem" refers to two individual theorems that pertain to formal mathematical systems with an arithmetical segment, where such a system is determined by its language, axioms, and rules of inference. With respect to the characteristics of such systems, the first incompleteness theorem speaks to the provability of arithmetic sentences within a specific system, and concerns the mathematical notions of consistency and completeness. Mathematically speaking, if a formal system is consistent, then it contains no contradictions; if the sentence A is provable in the system S using the axioms, language, and rules of S, then not-A is not provable within S. Conversely, if a formal system is inconsistent, then logical contradictions are provable by S, and for some sentence A, the sentences A and not-A are both provable within S. Or, as one commentator has it: "anything follows from a contradiction." Separately, if a formal system is complete, then that system permits the construction of proofs for all logical consequences following from the system's axioms and rules, and all consequences are decidable.
Taking these definitions and descriptions, we now state Gödel's first incompleteness theorem:
If the formal system S exhibits arithmetic and is consistent, then S is necessarily incomplete, and there exists an arithmetical sentence A in S that is undecidable in S; when S is consistent it is not also complete, and when S is complete it is not also consistent
(Note that this formulation is not categorical, but is composed to help us later locate connections between incompleteness and the mind. As noted, here we will focus on the content and significance of incompleteness, and not its technical aspects.)
The inability to decide the specific arithmetic sentence A pertains specifically to system S, while the characteristic "there exists an arithmetical sentence A in S that is undecidable in S" is applicable to formal mathematical systems exhibiting arithmetic. In connection to exhibiting arithmetic, the first theorem is apposite to any system that includes what Torkel Franzén has dubbed a "certain amount of arithmetic," meaning a formal system that includes sufficient arithmetical language, rules, and axioms to apply the first incompleteness theorem for that system. The phrase "certain amount of arithmetic" is somewhat esoteric, though necessarily so, as Franzén explains;
"if a property of natural numbers, such as being the sum of two primes, can be checked by a mechanical computation, then if a number n has that property, there is an elementary mathematical proof that n has the property. The 'certain amount of arithmetic' that a formal system S needs to encompass for the proof of the first incompleteness theorem to apply to S is precisely the arithmetic needed to substantiate this claim."
So, if the formal system S exhibits this certain amount of arithmetic, then by the first theorem we know S is incomplete with regard to its certain amount of arithmetic. Because the first theorem appertains to provability, it is notable that the certain amount of arithmetic does not need to be extensive or complicated, and the conclusion of the first theorem bears alike on systems that are elementary and on systems that are highly abstract.
Moving from completeness to consistency, the second incompleteness theorem speaks to the provability of the consistency of formal mathematical systems possessing a certain amount of arithmetic, and states:
If the formal system S exhibits arithmetic and is consistent, its consistency is not provable within S; it is not possible to prove ConS (the consistency of S) using the axioms, language, and rules of S
Like the first theorem, the second theorem describes a characteristic of formal mathematical systems that express arithmetic, and also like the first theorem, its references are not absolute, but relative to some specific system S. The statement "if the formal system S exhibits arithmetic and is consistent, its consistency is not provable within S" explicates the inability of proving ConS using system S in reference only to system S, and has no specific implications for other systems. It may be the case that ConS can be proved in another system, but the second theorem presents no insight regarding such prospects.
We thus observe that the title of the incompleteness theorem arises from its proof that the arithmetic segment of a formal system is incomplete and cannot provide decidability for all statements, and that the consistency of arithmetic cannot be shown arithmetically. These remarkable results were arrived at arithmetically, and this fact is central to understanding the generation and consequences of the theorem, which apply to formal systems in a general way.
In order to develop his proofs arithmetically and generally, Gödel contrived a method to enable a formal system to make statements such as ConS, which is a statement in S about S. This he achieved by the development of new mathematical innovations, Gödel sentences and the arithmetization of syntax, and by applying the basic logical implication that for the Gödel sentence G within the system S, if S is consistent then G is true. The Gödel sentence G is an arithmetical construct that can be calculated for arithmetical proofs and sentences, and because formal systems make statements about their own arithmetical constructs, Gödel sentences therefore permit formal systems to make statements about their own proofs and sentences. Accordingly, because Gödel sentences and the arithmetization of syntax are arithmetical, the incompleteness theorem applies generally to formal mathematical systems that possess the requisite arithmetic (Franzén's "certain amount of arithmetic"), whether a system does or does not include other mathematical or non-mathematical objects.
With respect to the consequences and implications of incompleteness, the theorem's conclusions are specifically about arithmetic, and the theorem does not explicitly state or conclude anything about a system's non-arithmetical or non-mathematical objects. This means the incompleteness theorem has no direct applicability outside of mathematics. Nevertheless, like all systems we develop, mathematically or otherwise, the theorem itself is a product of the human mind and must have some implications for the mind, if indirectly, and must have some implications for statements that are not strictly arithmetical or mathematical. What might some of those implications be, and how are they to be uncovered?
Reflecting on the content and conclusions of the incompleteness theorem as outlined above, we see the theorem suggests a number of questions and areas of inquiry, with respect to its implications for knowledge of the mind. In particular, if the mathematical systems we conceive cannot be both consistent and complete, or prove their own consistency, and no single system can provide decidability for all arithmetic, then what does it mean for humans to speak and think of the truthfulness of a mathematical proof? In this connection, what are the consequences of incompleteness for the mathematical frameworks that the mind is able to formalize, and what does incompleteness mean for human attempts to represent arithmetical knowledge using formal systems, and related attempts to convert formal systems in to machine-based representations of those systems? Does incompleteness imply the mathematical abilities of the mind exceed the abilities of (humanly built) systems and machines?
To develop a fundamental understanding of these issues, and uncover the relationship between incompleteness and its implications for human knowledge of the mind, we will discuss these questions individually, while also considering their interrelationships.
Beginning with the question of how humans should regard mathematical knowledge, it is instructive to express this question as a skeptical inquiry regarding the nature and outcomes of mathematics: are proofs generated using humanly defined mathematical systems to be taken as objectively and universally truthful, irrespective of humans, human thought, and intellectual structures? This is a popular and perennial question, not originally or specifically motivated by the incompleteness theorem, but the consequences of the incompleteness theorem do have a bearing on any answer.
The incompleteness theorem formalized that we can develop a consistent theory that proves the theory itself is not consistent, and that the consistency of a system is no guarantee the system will not prove false sentences. This was shown to be a general problem, and what this means is that it is right for skepticism to arise regarding the results of mathematical claims developed using even consistent systems. Thus, one of the most basic consequences of the incompleteness theorem for human knowledge of the mind is that we are rightly skeptical about the probity of knowledge arising within that particular subset of our knowledge that is encapsulated and expressed in the formal mathematical systems we conceive and work with.
Accepting this as inescapable, how should humans comport themselves towards formal mathematical proofs, and what does it mean for an arithmetical statement to be true? This line of inquiry also existed long before the advent of the incompleteness theorem, and while the incompleteness theorem by no means solves the question, it does strengthen our understanding that the soundness of an arithmetical proof imputes to that proof not the objective property that it is universally true irrespective of all qualifications, but rather, what we know is that the proof was generated in a logically conclusive manner using the language, axioms, and rules of inference of those systems that prove it. In this way, the incompleteness theorem assists in rounding out our notion of arithmetic truth, by showing us that particular proofs and theorems will always have truth and falsity relative to their systems, and not have truthfulness in an absolute, universal sense. This helps us better distinguish formal systemic truth from the notion of truthfulness for an arithmetical sentence, which is not relative (because in a non-relative way, uttering the "words 'the twin prime conjecture is true' is simply another way of saying exactly what the twin prime conjecture says. It is a mathematical statement, not a statement about ... any relation between language and a mathematical reality").
In contrast to the indirect observation that the incompleteness theorem "rounds out" our notion of arithmetic truth, one of the most important direct consequences of the incompleteness theorem pertains to David Hilbert's famous Program (which was a source of inspiration for the theorem's development). Where Hilbert sought to bring about the development of a concrete, consistent, formal system whose axioms, language, and rules determined the set of all arithmetical problems, Gödel's theorem proved that for any consistent formal system, there exists an undecidable sentence in the system. Therefore, regardless of the size or complexity of a system, no system can prove all statements concretely, and Hilbert's mathematical program as originally conceived is unattainable, because no single construction can expose and treat of all arithmetic. Thus, in addition to skepticism regarding the universal probity of mathematical statements conceived by the mind, the implication for the mind here is that it can never generate a formal definition that encompasses all human arithmetical knowledge.
Because we are unable to construct a single system that encompasses all human arithmetical knowledge, we therefore know our mathematical potential goes beyond our ability to formalize mathematics. To see this what this means, let us begin with the goal of maintaining systemic consistency, and consider the system S, to which we add the statement C; "S is consistent." S+C remains consistent. Now add C1; "S+C is consistent." S+C+C1 remains consistent; and so forth. Here we can continually construct simple systemic extensions with no end, by repeatedly adding to a consistent system the axiomatic statement that the existing system is consistent, thus yielding new systems that are consistent and determine more than prior systems. Proportionate to this demonstration, the implication of the incompleteness theorem for our knowledge of the mind is that its arithmetical knowledge is inexhaustible.
Inexhaustibility leads to questions about the limits of mathematical machines and computers, and has prompted some to claim that a direct consequence of incompleteness is that because our mathematical potential is inexhaustible and goes beyond our ability to formalize mathematics, we therefore know the human mind's arithmetical processing powers will always exceed those of computers, regardless of technological advancement. But does this follow from incompleteness? We know it possible to formalize our systems and represent them via computers and computer-based operations, and we also know we cannot fully capture our arithmetical abilities in a single system; but does this mean the human mind's arithmetical abilities exceed our computers?
If we consider Gödel sentences, then one fact we are certain of regarding these sentences is the formal implication that if a system S is consistent then its statements about its Gödel sentences are true. However, incompleteness does not tell us whether or not S is consistent. For this reason, the consequence of incompleteness for machine-based representations of human mathematical abilities is the familiar one: humans are unable to build a single computer-based representation of a formal system that solves every arithmetical problem, because we cannot define such a system. Thus, incompleteness does have implications for attempts at constructing machine-based representations of the human mind, but those implications do not include, or support the claim that the arithmetical ability of the human mind exceeds the arithmetical ability of humanly defined mathematical systems (be those systems computer-based or not). We cannot conclude from incompleteness that the mind's arithmetical processing powers will eternally exceed those of computers, regardless of technological developments.
Accepting the conclusions of the incompleteness theorem and reflecting on the discussion above, we can reasonably accept the following consequences of incompleteness for human knowledge of the mind: humans are right to approach their proofs and mathematical knowledge with some measure of skepticism, owing to the property of systemic relativity; our minds can not construct a formal definition that encompasses all humanly accessible arithmetical knowledge; the mind's arithmetical knowledge is inexhaustible; and, we are given no reason to believe the mind's arithmetical ability will permanently exceed the abilities of humanly defined systems, including computers. With this list of implications in hand, we discover while looking back to the opening comments of this paper the existence of a striking lack in the basic connections between incompleteness and the mind: madness.
Insofar as it is a product of the human mind, Gödel's incompleteness theorem is exceptional, and precisely because of this, when examining its content and implications the student of incompleteness must be careful, because the direct implications of the theorem are mathematical and metamathematical, and though the theorem's indirect implications do extend beyond their specific mathematical content, its non-mathematical implications are by no means simple to explicate. Notably, the truth of this is well supported by the existence of the book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, wherein the author's primary purpose is to disabuse readers of many inspiring, but patently incorrect interpretations of the incompleteness theorem inside and outside of mathematics, noting that much of the confusion regarding incompleteness stems from the fact that it is a commentary on systems.
Reflecting on systems and the conceptual development of mathematics, it appears we find in our history something of a human drive to systematize. In our written records, we observe repeated attempts to develop exact and exhaustive systems: from Euclid's Elements to Aristotle's Categories and Frege's Foundations of Arithmetic; from the Christian Bible to Bacon's Novum Organon and Hegel's Phenomenology of Spirit; and beyond. In these works the drive to systematize is manifest metaphysically, analytically, algebraically, and arithmetically; and, though it is true these works and their systems are products of the mind and must reflect the mind in some fashion, and though we also recognize with certainty that mathematics can be used to predict empirical phenomena, we also recognize that formally speaking, students of the philosophy of mathematics must be careful not to conflate mathematical concepts and conclusions with metaphysical precepts and suppositions. The content and methods of mathematics differs substantively from the content and methods of, for example, the physical and social sciences.
Gödel himself commented on the exigencies of human ideas, expression, and understanding, and noted; "The more I think about language, the more it amazes me that people ever understand each other at all."
Towards the end of understanding, and being careful not to misrepresent the basic implications of Gödel's incompleteness theorems for knowledge of the mind, we have accepted the systemic relativity of mathematical truth, the inability of total arithmetical formalization, mathematical inexhaustibility, and the fact that incompleteness does not mean the mind's arithmetical abilities will continue to exceed those of computers. Expressly, what we do not accept is any suggestion, however oblique, that the content of the incompleteness theorem somehow drove Kurt Gödel mad, or that the content of the theorem might be used to explain madness.
While incompleteness does have implications for human knowledge of the mind, we must respond to all claims that systematize the mathematics of madness with an elementary inquiry: if powerful mathematical insight induces madness, why is there not an endless profusion of insane mathematicians and math students? If the content of the incompleteness theorem drove Kurt Gödel mad, then surely others who have worked with the theorem -- or the theories of Cantor, Boltzmann, and Turing -- should have been driven to madness as well. Here we find ourselves far removed from the realm of mathematical thought, and, as a consequence of our analysis above, we understand that any discussion of madness is far removed from discussion of Gödel's incompleteness theorem.
Part of the series: UWO