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I've read Godel, Escher, Bach, and I've read this. It's a very nice explanation. But everywhere I see, Godel's theorem is touted as some kind of deep philosoph
by hvis 6y ago
I've read Godel, Escher, Bach, and I've read this. It's a very nice explanation.
But everywhere I see, Godel's theorem is touted as some kind of deep philosophical insight, whereas from what I understand, informally it could be rephrased as "if you have a usable language for mathematical proofs, some phrases in that language must be neither true nor false (i.e. nonsensical)".
Nonsensical phrases in our human languages in nothing new, so the conclusion becomes that much less exciting. Though I'm sure it's valuable for fundamental math theory.
- nmg 6y agoTo me reading this article it was striking to consider the relationship between language and (mathematicsl/logical) truth this way. In my mind it correlates with the quantum phenomenon that observing something fundamentally alters its state.
- nmadden 6y agoIt’s not so much that they’re neither true nor false (they’re not paradoxes), but that they can be true in some interpretations and false in others. I wrote up some notes about this here: https://neilmadden.blog/2020/11/17/some-incomplete-thoughts-about-godel/ https://neilmadden.blog/2020/11/17/some-incomplete-thoughts-...
- hvis 6y agoWasn't that proof using a paradox, though? Then that basically really proves the existence of paradoxes. It doesn't disprove the existence of all other kinds of statements, of course. A statement 'x == 1' might be contextual (with a free variable x), but I think the theory is talking more about statements like for all x (x + 1)^ 2 == x^2 + 2*x + 1 which is true and does not depend on x. Contrast that with for all x x == 1 which is obviously false. But the proof outlined in the article constructed a statement with a free variable, where a substitution with a certain value leads to a statement which is neither true, nor false. That means that the general statement in the proof's example is neither true nor false either. Sorry about the awkward terminology, BTW, my engineering degree was a decade ago and in a language other than English.
- nmadden 6y agoNo, the statements in Gödel’s proofs are not paradoxes. Substitutions of variables are not the same as interpretations. The interpretation tells you what the nonlogical symbols like *, +, ^, 1, 2, 3 etc mean.
- praptak 6y agoNo, it is not about nonsensical phrases. Godel proved that there are some sentences that are true, but they have no proof in the system.
- ProfHewitt 6y agoHi Hvis! Inferentially undecidable sentences of Russell's Principia Mathematica are not nonsensical. However, they are obscure because no particular example is known because no one knows how to present them constructively.