We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths.
As I read it.
I see nothing that leads me to believe that we're capable of grasping the complete truth in mathematics.We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths.
As I read it.
I was responding to this specific quote from Yourgrau (my underlines):
... The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers.
So, by this argument, we'd have to grasp the complete truth of mathematics (whatever that might mean) in order to conclude that "we, or our minds, are not machines or computers." I was calling into question the likelihood of our being able to grasp the complete truth of mathematics. And, if we can't, it no longer follows (from this argument) that "we, or our minds, are not machines or computers".
Good to hear from you, bud...
[headsonpikes:] We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths. As I read it.
Too many people make the mistake of concluding that if human minds can grasp more theorems than a consistent formal system like mathematics, then our minds must not be "computer-like" (i.e. not a formal system, or not deterministic).
This is a fallacy.
Here's what I wrote on that issue in my review of Robert Penrose's, "The Emperor's New Mind":
Additionally, it's not encouraging that even when arguing a point in his own field of mathematics, he miapplies it. He invokes Godel's Incompleteness Theorem in support of his speculations about mind, but makes the same mistaken made by countless metaphysicists before him -- and rebutted by countless other people already.It only demonstrates that the human mind is not a *consistent* formal system -- and even that conclusion holds only if the human mind is demonstrably "complete", which is certainly an unwarranted presumption.Godel's Theorem states that a formal system cannot be simultaneously consistent and complete. Penrose makes two mistakes when he applies this to mind:
1. He equates "formal system" with "deterministic system", but they are not synonyms. All formal systems are deterministic, but not all deterministic systems operate in the manner of formal systems. [i.e., the limitation of formal systems is not necessarily a limitation on *all* types of deterministic systems.]
2. Penrose presumes that since human minds can arrive at conclusions which formal systems can not, that therefore we are not subject to Godel's Incompleteness Theorem, and thus our minds are not formal/deterministic systems. Penrose forgets that the more obvious reason we can do so is contained in the flip-side of Godel's two opposing conditions -- the human mind is not *consistent*. That is, it is capable of reaching conclusions which are wrong and/or contradict other conclusions. Godel's Theorem is *only* a limitation on *consistent* formal systems -- that is, those which never produce contradictory results.
Needless to say, the human mind is not such a system. We're quite capable of contradicting ourselves, in a logical sense. We sacrifice logical consistency (and therefore rigid accuracy) for flexibility. This does indeed allow us to recognize truths that formal systems are "blind" to -- but at the expense of being able to make mistakes as well. Formal systems may be limited by the Godel Incompleteness Theorem -- but at the same time they're capable of producing *only* correct proofs. There are trade-offs and advantages either way.
In short, the human mind can exceed the abilities of formal systems (in *some* regards) not because the mind is nondeterministic, but because it uses "fuzzy" logic that isn't guaranteed to be always accurate. Win some, lose some.
Godel's Incompleteness Theorem is no support for the hypothesis that the human mind cannot be, at its foundation, some sort of formal or deterministic system.
It's also important to note that a deterministic computer which just mechanically enumerated all postulates would likewise be complete, albeit at the same cost -- inconsistency. So would one which enumerated all postulates, then weeded out some of the more obviously false (Godel's theorem shows that it can't weed out *all* of them). Such a computer, if self-aware, might likewise try to argue that since it can "find" more truths than a formal system, it's non-deterministic, but again that would be incorrect, for the reasons described above (and obviously so in this case, since the computer *is* acting deterministically).