Editing Gödel's incompleteness theorems (section)

=== 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>