Editing Gödel's incompleteness theorems (section)

== Discussion and implications ==
The incompleteness results affect the [[philosophy of mathematics]], particularly versions of [[Symbolic Logic|formalism]], which use a single system of formal logic to define their principles.

=== Consequences for logicism and Hilbert's second problem ===

The incompleteness theorem is sometimes thought to have severe consequences for the program of [[logicism]] proposed by [[Gottlob Frege]] and [[Bertrand Russell]], which aimed to define the natural numbers in terms of logic.{{sfn|Hellman|1981|pp=451–468}} [[Bob Hale (philosopher)|Bob Hale]] and [[Crispin Wright]] argue that it is not a problem for logicism because the incompleteness theorems apply equally to first-order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to [[David Hilbert]]'s [[Hilbert's second problem|second problem]], which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "[[Hilbert's second problem#Modern viewpoints on the status of the problem|Modern viewpoints on the status of the problem]]").

=== Minds and machines ===
{{Main|Mechanism (philosophy)#Gödelian arguments}}

Authors including the philosopher [[John Lucas (philosopher)|J. R. Lucas]] and physicist [[Roger Penrose]] have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a [[Turing machine]], or by the [[Church–Turing thesis]], any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.

{{harvtxt|Putnam|1960}} suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.{{sfn|Putnam|1960}}

{{harvtxt|Wigderson|2010}} has proposed that the concept of mathematical "knowability" should be based on [[Computational complexity theory|computational complexity]] rather than logical decidability.  He writes that "when ''knowability'' is interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."{{sfn|Wigderson|2010}}

[[Douglas Hofstadter]], in his books ''[[Gödel, Escher, Bach]]'' and ''[[I Am a Strange Loop]]'', cites Gödel's theorems as an example of what he calls a ''strange loop'', a hierarchical, self-referential structure existing within an axiomatic formal system. He argues that this is the same kind of structure that gives rise to consciousness, the sense of "I", in the human mind. While the self-reference in Gödel's theorem comes from the Gödel sentence asserting its unprovability within the formal system of Principia Mathematica, the self-reference in the human mind comes from how the brain abstracts and categorises stimuli into "symbols", or groups of neurons which respond to concepts, in what is effectively also a formal system, eventually giving rise to symbols modeling the concept of the very entity doing the perception. Hofstadter argues that a strange loop in a sufficiently complex formal system can give rise to a "downward" or "upside-down" causality, a situation in which the normal hierarchy of cause-and-effect is flipped upside-down. In the case of Gödel's theorem, this manifests, in short, as the following:
<blockquote> Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false.{{sfn|Hofstadter|2007}} </blockquote>

In the case of the mind, a far more complex formal system, this "downward causality" manifests, in Hofstadter's view, as the ineffable human instinct that the causality of our minds lies on the high level of desires, concepts, personalities, thoughts, and ideas, rather than on the low level of interactions between neurons or even fundamental particles, even though according to physics the latter seems to possess the causal power.
<blockquote> There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside.{{sfn|Hofstadter|2007}} </blockquote>

=== Paraconsistent logic ===

Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of [[paraconsistent logic]] and of inherently contradictory statements (''[[dialetheia]]''). {{harvs|last=Priest|year=1984|year2=2006|txt}} argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for [[dialetheism]].{{sfnm
 | 1a1 = Priest | 1y = 1984
 | 2a1 = Priest | 2y = 2006
}} The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system.{{sfn|Priest|2006|p=47}} {{harvtxt|Shapiro|2002}} gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.{{sfn|Shapiro|2002}}

=== Appeals to the incompleteness theorems in other fields ===

Appeals and analogies are sometimes made to the incompleteness of theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including {{harvtxt|Franzén|2005}}, {{harvtxt|Raatikainen|2005}}, {{harvtxt|Sokal|Bricmont|1999}}; and {{harvtxt|Stangroom|Benson|2006}}.{{sfnm
 | 1a1 = Franzén | 1y = 2005
 | 2a1 = Raatikainen | 2y = 2005
 | 3a1 = Sokal | 3a2 = Bricmont | 3y = 1999
 | 4a1 = Stangroom | 4a2 = Benson | 4y = 2006
}}
{{harvtxt|Sokal|Bricmont|1999}} and {{harvtxt|Stangroom|Benson|2006}}, for example, quote from [[Rebecca Goldstein]]'s comments on the disparity between Gödel's avowed [[Mathematical Platonism|Platonism]] and the [[anti-realist]] uses to which his ideas are sometimes put. {{harvtxt|Sokal|Bricmont|1999}} criticize [[Régis Debray]]'s invocation of the theorem in the context of sociology; Debray has defended this use as metaphorical (ibid.).{{sfnm
 | 1a1 = Sokal | 1a2 = Bricmont | 1y = 1999
 | 2a1 = Stangroom | 2a2 = Benson | 2y = 2006 | 2p = 10
 | 3a1 = Sokal | 3a2 = Bricmont | 3y = 1999 | 3p = 187
}}