I Am a Strange Loop -- Chapters 10-12
I'll skip chapter 9 and move on to chapter 10. Assume Theorem Z is the consequence of Theorems X and Y, where x,y, and z are the corresponding Goedel numbers. Then the relation between x,y, and z mirrors that between X, Y, and Z. This is how Hofstadter sums up the correpondence:
"...if x were the number corresponding to theorem X and y were the number corresponding to theorem Y, then z would "miraculously" turn out to be the number corresponding theorem Z."(Hofstadter pg. 163 Nookbook).
He goes on to explain Goedel numbering very well. He explains well the importance of building up the correspondence between numbers and formulae in PM recursively until we have numerical relationships representing provability. He does an excellent job explaining Goedel's generation of an unprovable formula. He has a nice digression on Quine and Berry which is worth reading.
I'm going to skip over chapter 11 and move on to Chapter 12. Here he talks a lot about the consistency of PM and how that implies that the undecidable proposition must be true. He also shows how if the proposition were provable it would be false: recall that the proposition corresponds to the relation that holds when it is UNprovable. He concludes:
"In other words KG[the undecidable statement] is unprovable not only although it is true, but worse yet, because it is true."(Hofstadter pg 198, Nookbook).
This discussion is very nice. He pauses here on how perverse this situation is: a proposition is unprovable because it is true.
Now comes the crucial point for Hofstadter:
"PM is rich enough to be able to turn around and point at itself, like a television camera pointing at the screen to which it is sending its image. If you make a good enough TV system, this looping-back ability is inevitable. And the higher the system's resolution is, the more faithful the image is."(Hofstader pp. 199-200 Nookbook).
He then mentions something that is worth mentioning, that there are an infinite number of ways of numbering PM and there are distinct undecidable propositions for all of them! That is, there are an infinite number of undecidable propositions in PM.
Now, here is where the Hofstadter and I start to part company: Downward Causality in Mathematics. Here is what he says:
"It reveals the stunning fact tht the formula's hidden meaning may have a peculiar kind of "downward" causal power, determining the formula's truth or falsity...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."(Hofstadter pg. 203, Nookbook)
It is NOT the formula's hidden meaning that has causality. WE understand the meaning, the mapping via Goedel numbers provided the proof the undecidable proposition is true assuming consistency, which remember we can't prove from within PM. The hidden meaning is not some independent thing that has causal power all on its own; I know that Hofstadter will say " it's an epiphenomena and they can have causal power", but I think he's wrong here. All the formal system has is the rules of syntax. I agree that it is (much) easier to think of the meaning of the proposition than building up from the bottom. But in the end the proof of the incompleteness theorem is quite mechanical. He derives contradictions from assuming either the undecidable proposition or its negation. There is nothing spooky going on in the proof. Hofstadter has semantics popping out of the syntax all on its own like Athena from the head of Zeus. NO! I have a bad feeling now that consciousness will pop out of unconscious molecules in some analogous way -- I hope not, but we shall see. Well, I suspect things will not be put with that much clarity
"...if x were the number corresponding to theorem X and y were the number corresponding to theorem Y, then z would "miraculously" turn out to be the number corresponding theorem Z."(Hofstadter pg. 163 Nookbook).
He goes on to explain Goedel numbering very well. He explains well the importance of building up the correspondence between numbers and formulae in PM recursively until we have numerical relationships representing provability. He does an excellent job explaining Goedel's generation of an unprovable formula. He has a nice digression on Quine and Berry which is worth reading.
I'm going to skip over chapter 11 and move on to Chapter 12. Here he talks a lot about the consistency of PM and how that implies that the undecidable proposition must be true. He also shows how if the proposition were provable it would be false: recall that the proposition corresponds to the relation that holds when it is UNprovable. He concludes:
"In other words KG[the undecidable statement] is unprovable not only although it is true, but worse yet, because it is true."(Hofstadter pg 198, Nookbook).
This discussion is very nice. He pauses here on how perverse this situation is: a proposition is unprovable because it is true.
Now comes the crucial point for Hofstadter:
"PM is rich enough to be able to turn around and point at itself, like a television camera pointing at the screen to which it is sending its image. If you make a good enough TV system, this looping-back ability is inevitable. And the higher the system's resolution is, the more faithful the image is."(Hofstader pp. 199-200 Nookbook).
He then mentions something that is worth mentioning, that there are an infinite number of ways of numbering PM and there are distinct undecidable propositions for all of them! That is, there are an infinite number of undecidable propositions in PM.
Now, here is where the Hofstadter and I start to part company: Downward Causality in Mathematics. Here is what he says:
"It reveals the stunning fact tht the formula's hidden meaning may have a peculiar kind of "downward" causal power, determining the formula's truth or falsity...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."(Hofstadter pg. 203, Nookbook)
It is NOT the formula's hidden meaning that has causality. WE understand the meaning, the mapping via Goedel numbers provided the proof the undecidable proposition is true assuming consistency, which remember we can't prove from within PM. The hidden meaning is not some independent thing that has causal power all on its own; I know that Hofstadter will say " it's an epiphenomena and they can have causal power", but I think he's wrong here. All the formal system has is the rules of syntax. I agree that it is (much) easier to think of the meaning of the proposition than building up from the bottom. But in the end the proof of the incompleteness theorem is quite mechanical. He derives contradictions from assuming either the undecidable proposition or its negation. There is nothing spooky going on in the proof. Hofstadter has semantics popping out of the syntax all on its own like Athena from the head of Zeus. NO! I have a bad feeling now that consciousness will pop out of unconscious molecules in some analogous way -- I hope not, but we shall see. Well, I suspect things will not be put with that much clarity
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