"The time has been, That, when the brains were out, the man would die, And there an end; but now they rise again, With twenty mortal murders on their crowns, And push us from our stools"

-- Macbeth

The main character, the uber-hacker Case, and his girlfriend, Molly, who has Wolverine blades that shoot out from her knuckles, break into a conglomerate to steal the ROM construct of Case's dead mentor. They need the assistance of this mentor for the performance of their real job, helping an artifical intelligence named Wintermute(named after the translator of the Nag Hammadi Library) merge with another artificial intelligence named Neuromancer(Neuro, Romancer, and Necromancer jammed together).

The nickname of the mentor's construct is Flatline; his real name is Dixie McCoy:

"It was disturbing to think of the Flatline as a construct, a hardwired ROM cassette replicating a dead man's skills, obsessions, knee-jerk responses...."(N, 72)

Below is a conversation between Case and the ROM construct:

"He[Case] coughed. "Dix? McCoy? That you man?" His throat was tight.

"Hey, bro," said a directionless voice.

"It's Case, man, Remember?"

"Miami joeboy, quick study."

"What's the last thing you remember before I spoke to you, Dix?"

"Nothin'."

"Hang on." He disconnected the construct. The presence was gone. He reconnected it. "Dix?

Who am I?"

"You got me hung, Jack....

"Remember being here, a second ago?"

"No."

"Know how a ROM personality matrix works?"

....

"Okay, Dix. You ARE a ROM construct. Got me?"

"If you say so."

"Who are you?"

"Case."

"Miami,"

"joeboy, quick study."

[a joeboy is some kind of assistant to an underground mob/hacker boss type]

Notice that in the middle of the conversation Case disconnects the construct and then, I think as a kind of test, restarts the conversation. The construct remembers nothing from the previous instantiation. I'm guessing that because it's ROM, nothing can be added to the memory; thus this personality is frozen in the state it was when it was recorded. According to wikipedia, Neuromancer is able to construct RAM personalities, which means the personalities can grow.

Also notice that Flatline responds with "If you say so" when told he is a ROM construct. I take this to mean that Flatline is actually floating in mental space, unaware of where it is physically; it hovers, unable to construct new narratives about its experience. Then it can just be shut off. This is extremely creepy and well done.

I will think about the connection between the artificial intelligences and Gnosticism in the meantime. Apparently this is also an homage to a character in Phillip K. Dick's VALIS books: http://en.wikipedia.org/wiki/VALIS

Later in the book we will see the threat a merger between Wintermute and Neuromancer is thought to pose.

## Thursday, June 30, 2011

## Tuesday, June 28, 2011

### Cyberpunk: Neuromancer -- Entry 2

In Neuromancer we find out that the "Sprawl", the name given to a Gibson trilogy of books, is the Boston-Atlanta Metropolitan Axis, or BAMA. Gibson portrays the new world in terms of a map color-coded based on how much data exchange is occurring. So much data is being transmitted that you only begin seeing differences when you get above 100 million megabytes, which for today's metropolitan areas is nothing. Gibson also has the idea that there could be a black market for as little as 3 megabytes of RAM. There's no way he could have known in 1984 that RAM would be so cheap now.

Suspicion of cities as centers of moral turpitude goes back at least to Genesis. What's the deal here? The simple, rural life is contrasted with the complex, corrupt life in the population centers; city ways are not country ways. But what happens when the city takes over? Or what happens when the values of the city are exported to the country via instant communication or the Internet?

City ways separate us from organic origins and yield fake life, as in the electronic recording(a construct) of a dead hacker's mind. Now, is the recording actually alive? What kind of life is life in cyberspace? It's a hallucination of life, like the old brain in a vat. What I find wonderful is that Gibson says the matrix(cyberspace) has its origin in arcade games, games where as a kid I would get sucked into an alternate reality -- I had a similar obsession with non-electronic games like old Dungeons and Dragons, so I don't think escape has to be electronic but it is in this novel. The mind gets sucked in and eventually can be copied, that is, turned into what Gibson call a "construct", into the game itself; one becomes, as it were, "one" with the matrix, even surviving death as an electronic copy.

While Case can no longer experience the high from drugs because his pancreas has been replaced, he can still experience ecstasy in cyberspace. This all has mystical overtones, when he is in cyberspace:

"...Symbols, figures, faces, a blurred, fragmented mandala of visual information.

Please, he prayed,

And later:

Inner eye opening to the stepped scarlet pyramid of the Eastern Seaboard Fission Authority burning beyond the green cubes of Mitsubishi Bank of America, and high and very far away he saw the spiral arms of military systems, forever beyond his reach."(N, pg.50)

Gibson might as well be describing a peyote-induced vision.

In "reality" Case is really in his loft, hooked up to the computer:

"And somewhere he was laughing, in a white-painted loft, distant fingers caressing the deck, tears of release streaking his face"(N, pg. 50)

Suspicion of cities as centers of moral turpitude goes back at least to Genesis. What's the deal here? The simple, rural life is contrasted with the complex, corrupt life in the population centers; city ways are not country ways. But what happens when the city takes over? Or what happens when the values of the city are exported to the country via instant communication or the Internet?

City ways separate us from organic origins and yield fake life, as in the electronic recording(a construct) of a dead hacker's mind. Now, is the recording actually alive? What kind of life is life in cyberspace? It's a hallucination of life, like the old brain in a vat. What I find wonderful is that Gibson says the matrix(cyberspace) has its origin in arcade games, games where as a kid I would get sucked into an alternate reality -- I had a similar obsession with non-electronic games like old Dungeons and Dragons, so I don't think escape has to be electronic but it is in this novel. The mind gets sucked in and eventually can be copied, that is, turned into what Gibson call a "construct", into the game itself; one becomes, as it were, "one" with the matrix, even surviving death as an electronic copy.

While Case can no longer experience the high from drugs because his pancreas has been replaced, he can still experience ecstasy in cyberspace. This all has mystical overtones, when he is in cyberspace:

"...Symbols, figures, faces, a blurred, fragmented mandala of visual information.

Please, he prayed,

*now*"(N, pg*.*50)And later:

Inner eye opening to the stepped scarlet pyramid of the Eastern Seaboard Fission Authority burning beyond the green cubes of Mitsubishi Bank of America, and high and very far away he saw the spiral arms of military systems, forever beyond his reach."(N, pg.50)

Gibson might as well be describing a peyote-induced vision.

In "reality" Case is really in his loft, hooked up to the computer:

"And somewhere he was laughing, in a white-painted loft, distant fingers caressing the deck, tears of release streaking his face"(N, pg. 50)

## Sunday, June 26, 2011

### Cyberpunk: Neuromancer -- Chapter 1

I've finished reading the first chapter through a couple of times and am thinking hard about it. It seems clear that there is a lot of nihilism in cyberpunk. Clearly traditional values, and the quest for meaning generally, have been discarded: there is nothing but the super-fast movement of culture and technology, a sense of disorientation, a definite, and by now very trite, noir sensibility, a sense of humanity overwhelmed by change and super-smart machines, machines which have become conscious, leading us to the brink of the "technological singularity"(see the wikipedia page) that leads to the end of human history, a time machines will then recursively outpace us, machines will drive history, not us. In this way, cyberpunk is thoroughly postmodern, but in the way that we all have to face up to, not in some purely academic way where you are not allowed to have subjects, verbs, and objects in sentences unless you cross them out.

The first line of the novel,

"The sky above the port was the color of television, tuned to a dead channel"(N, pg.7),

obviously reminds me of the first line of the Love Song of J. Alfred Prufrock, by T.S. Eliot:

'Let us go then, you and I, when the evening is spread out against the sky, like a patient etherized upon a table"

In The Wasteland, Eliot refers to the "Unreal City", and Chiba City in the novel is real, hyper-real, and surreal -- one of the characters even has a melted Dali clock on the wall to drive this home. The hyper-real cyberspace is a merging of human beings with computers, consciousness is separated from the body. Man, this sounds like the books I've been reading; check out my last section of entries on

"...jacked into a custom cyberspace deck that projected his disembodied consciousness into the consensual hallucination that was the matrix." (N pg. 9)

"For Case, who'd lived for the bodiless exultation of cyberspace, it was the Fall...The body was meat. Case fell into the prison of his own flesh."(N, pg.9)

The desire to escape to cyberspace is balanced perfectly by the thanatos embodied by the city itself:

"Ninsei wore him down until the street itself came to seem the externalization of some death wish, some secret poison he hadn't known he carried."(N, pg. 10)

The beautiful sentence below gives the sense of nihilism caused by rapid change in a heartless society dominated by massive corporations and organized crime:

"Night City was like a deranged experiment in social Darwinism, designed by a bored researcher who kept one thumb permanently on the fast-forward button."(N, pg. 10)

There are no humane values, just technology and profit to be had. The world is what happens when an Ayn Randian philosophy has a chance to move forward. Even the hero of our story seems self-centered; he is more an antihero than anything else.

Case speculates that this region was allowed to exist so that there would be a place where technology could have free reign; technology could mutate in the same way biological entities do so that it can adapt. It is an interesting thought that technology should be allowed to compete with itself so that the better technologies survive. Human beings are then the vehicles, the meat sacs carrying the technologies that are really the point of the evolution. This is where one could speculate that the "singularity" has approached and human evolution is replaced with machine evolution; human beings are the pawns in this game.

Looking at shurikins in a shop window Case thinks:

"...it came to Case that these were the stars under which he voyaged, his destiny spelled out in a constellation of cheap chrome."(N,15)

The stars that guide the world are ultimately the impersonal forces of technology and the dark forces of huge corporations. Can he have anything like freedom in this context? Only, it seems, through more violence.

He then draws a parallel between the artificial and the biological in the remarkable passage below. Notice the "Word became flesh and dwelt among us" from The Bible mutates into "data made flesh":[I'm still thinking about this one, when I have got some stuff thought out I'll come back to it]

"Get just wasted enough, find yourself in some desperate but strangely arbitrary kind of trouble, and it was possible to see Ninsei as a field of data, the way the matrix had once reminded him of proteins linking to distinguish cell specialties. Then you could throw yourself into a high speed drift and skid, totally engaged but set apart from it all, and all around you the dance of the biz, information interacting, data made flesh in the mazes of the black market..."(N,19)

The first line of the novel,

"The sky above the port was the color of television, tuned to a dead channel"(N, pg.7),

obviously reminds me of the first line of the Love Song of J. Alfred Prufrock, by T.S. Eliot:

'Let us go then, you and I, when the evening is spread out against the sky, like a patient etherized upon a table"

In The Wasteland, Eliot refers to the "Unreal City", and Chiba City in the novel is real, hyper-real, and surreal -- one of the characters even has a melted Dali clock on the wall to drive this home. The hyper-real cyberspace is a merging of human beings with computers, consciousness is separated from the body. Man, this sounds like the books I've been reading; check out my last section of entries on

*I Am A Strange Loop.*Consider Chase's description of the cyberspace experience:"...jacked into a custom cyberspace deck that projected his disembodied consciousness into the consensual hallucination that was the matrix." (N pg. 9)

*The passage below shows the distinction between disembodied mind and a life dominated by the flesh. We see that cyberspace is like heaven and the world is like hell:*"For Case, who'd lived for the bodiless exultation of cyberspace, it was the Fall...The body was meat. Case fell into the prison of his own flesh."(N, pg.9)

The desire to escape to cyberspace is balanced perfectly by the thanatos embodied by the city itself:

"Ninsei wore him down until the street itself came to seem the externalization of some death wish, some secret poison he hadn't known he carried."(N, pg. 10)

The beautiful sentence below gives the sense of nihilism caused by rapid change in a heartless society dominated by massive corporations and organized crime:

"Night City was like a deranged experiment in social Darwinism, designed by a bored researcher who kept one thumb permanently on the fast-forward button."(N, pg. 10)

There are no humane values, just technology and profit to be had. The world is what happens when an Ayn Randian philosophy has a chance to move forward. Even the hero of our story seems self-centered; he is more an antihero than anything else.

Case speculates that this region was allowed to exist so that there would be a place where technology could have free reign; technology could mutate in the same way biological entities do so that it can adapt. It is an interesting thought that technology should be allowed to compete with itself so that the better technologies survive. Human beings are then the vehicles, the meat sacs carrying the technologies that are really the point of the evolution. This is where one could speculate that the "singularity" has approached and human evolution is replaced with machine evolution; human beings are the pawns in this game.

Looking at shurikins in a shop window Case thinks:

"...it came to Case that these were the stars under which he voyaged, his destiny spelled out in a constellation of cheap chrome."(N,15)

The stars that guide the world are ultimately the impersonal forces of technology and the dark forces of huge corporations. Can he have anything like freedom in this context? Only, it seems, through more violence.

He then draws a parallel between the artificial and the biological in the remarkable passage below. Notice the "Word became flesh and dwelt among us" from The Bible mutates into "data made flesh":[I'm still thinking about this one, when I have got some stuff thought out I'll come back to it]

"Get just wasted enough, find yourself in some desperate but strangely arbitrary kind of trouble, and it was possible to see Ninsei as a field of data, the way the matrix had once reminded him of proteins linking to distinguish cell specialties. Then you could throw yourself into a high speed drift and skid, totally engaged but set apart from it all, and all around you the dance of the biz, information interacting, data made flesh in the mazes of the black market..."(N,19)

## Saturday, June 25, 2011

### Cyberpunk and Post Cyberpunk

For my next project, I will read

In the meantime check out this website from Washington State University Profesor Paul Brians:

http://public.wsu.edu/~brians/science_fiction/neuromancer.html

Note that I will not be answering any of his study questions, at least not intentionally; I will leave that to his students or interested readers.

*Neuromancer*, by William Gibson, an anthology of post cyberpunk stories, some Neal Stephenson, and look at the development from cyberpunk, post cyberpunk, and maybe even post post cyberpunk. It turns out that many of the issues raised in the couple of books I've reviewed, especially the Hofstadter, bear directly on the worlds created by these various books.In the meantime check out this website from Washington State University Profesor Paul Brians:

http://public.wsu.edu/~brians/science_fiction/neuromancer.html

Note that I will not be answering any of his study questions, at least not intentionally; I will leave that to his students or interested readers.

## Friday, June 24, 2011

### I Am a Strange Loop -- Conclusion

In an interesting twist, in chapter 22 Hofstadter reveals that David Chalmers was a doctoral student of his. It so happens that Chalmers has made a career out of taking positions opposed to that of his former advisor. Chalmers is a champion of philosophical zombie thought experiments. Chalmers lectures on the notion that an unconscious copy of each of us is conceivable: it may take an alternate universe, but it is conceivable. As a result, there is a gap between the physical and mental. Hofstadter does a lot of poking good-natured fun at his former student, but he never adduces a single argument of any power.

In the end,

In the end,

*I am a strange loop,*as interesting a read as it has been, has not convinced me ot any of Hofstadter's distinctive positions; I have to say I agree more with John Searle. In the end, and here I disagree with John Searle, I am pessimistic we will be able understand consciousness beyond the level of correlation with physical substrates; I suspect it is a limitation of our condition.## Wednesday, June 22, 2011

### I Am a Strange Loop -- a skip all the way through to Chapter 18

I am going to skip to chapter 18 to one of the most distinctive of Hofstadter's ideas. In the intervening chapters Hofstadter reiterates his notion of the strange-loop, the necessity to change the locus of causality to symbols rather than to the "chemical squirting" substrate of the brain, a theory with which I have some disagreement, and contains at least one mention of his nemesis, John Searle. In chapter 18, though, his theory of the self takes on a really interesting turn, as a subtitle has it: "I Host and am Hosted by Others(Hofstadter, pg. 301).

I'll let him speak for himself here:

"...the idea I am proposing here is that since a normal adult human brain is a representationally universal "machine", and since humans are social beings, an adult brain is the locus not only of

Well, that

If I am reading him aright, and maybe I'm not, he's saying that the actual consciousnesses of other human beings inhabit us to some degree -- like we're possessed. The representative powers of the brain allows it to mimic some portion of the strange loop of other people; hence their consciousness inhabits us. We are therefore conscious in multiple places at once.

He says later:

"That's all I'm claiming -- that there is a blur. That some of what happens in other brains gets copied, albeit coarse-grainedly, inside the brain of "Number One", and that the closer two brains are to each other emotionally, the more stuff gets copied back and forth from one to the other, and the more faithful the copies are. There's no claim that the act of copying is simultaneous or perfect or total -- just that each person lives

That's all he's claiming...

Well, I have to say that I disagree with him here, rather extremely, but it is a really cool idea. Really.

I'll let him speak for himself here:

"...the idea I am proposing here is that since a normal adult human brain is a representationally universal "machine", and since humans are social beings, an adult brain is the locus not only of

*one*strange loop constituting the identity of the primary person associated with that brain, but of*many*strange-loop patterns that are coarse-grained copies of the primary strange loops housed in other brains. Thus, brain 1 consists of strange loops 1, 2, 3, and so forth, each with its own level of detail. ... Every normal adult human soul is housed in many brains to varying degrees of fidelity, and therefore every human consciousness or "I" lives at once in a collection of different brains, to different extents."(Hofstadter pg.301 Nookbook)Well, that

*is*interesting. He goes on to say: "The interpenetration of souls is an inevitable consequence of the power of the representationally universal machines that our brains are."(Hofstadter pg. 309)If I am reading him aright, and maybe I'm not, he's saying that the actual consciousnesses of other human beings inhabit us to some degree -- like we're possessed. The representative powers of the brain allows it to mimic some portion of the strange loop of other people; hence their consciousness inhabits us. We are therefore conscious in multiple places at once.

He says later:

"That's all I'm claiming -- that there is a blur. That some of what happens in other brains gets copied, albeit coarse-grainedly, inside the brain of "Number One", and that the closer two brains are to each other emotionally, the more stuff gets copied back and forth from one to the other, and the more faithful the copies are. There's no claim that the act of copying is simultaneous or perfect or total -- just that each person lives

*partially*in the brain of the other..."(Hofstadter, pg 313)That's all he's claiming...

Well, I have to say that I disagree with him here, rather extremely, but it is a really cool idea. Really.

### I Am a Strange Loop -- and a strange digression on Foucault--Chapter 13

In Chapter 13, Hofstadter focuses on the notion of the "I" as a large structure of neural processes within the brain best represented symbolically, recall the much earlier description of large scale structures being causally effective as opposed to only crediting micro-processes with causal power. He talks about the sense of self as extensible through our past and providing a sense of unity to our experience, prompting, Kant to think of it as the "transcendental unity of apperception". He says:

"Since we perceive not particles interacting but macroscopic patterns in which certain thing s push other things around with a blurry causality, and since the Grand Pusher in and of our bodies is our "I", and since our bodies push the rest of the world around, we are left with no choice but to conclude that the "I" is where the causality buck stops."(Hofstadter, pg. 217 Nookbook).

This "I" gains structure as we get older:

"We begin life with the most elementary sorts of feedback about ourselves, which stimulate us to formulate categories for our most obvious body parts, and building on this basic pedestal, we soon develop a sense of our bodies as flexible physical objects. In the meantime, as we receive rewards for various actions and punishments for others, we being to develop a more abstract sense of "good" and "bad", as well as notions of guilt and pride, and our sense of ourselves as abstract entities that have the power to decide to make things happen ... begins to take root."(Hofstadter, pg. 218, Nookbook)

This remarkable passage bears a striking resemblance to some ideas of Michel Foucault in

"This real, noncorporal soul, is not a substance; it is the element in which are articulated the effects of a certain type of power and the reference of a certain type of knowledge, the machinery by which the power relations give rise to a possible corpus of knowledge, and knowledge extends and reinforces the effects of this power. On this reality-reference, various concepts have been constructed and domains of analysis carried out: psyche, subjectivity, personality, consciousness etc.; on it have been built scientific techniques and discourses, and the moral claims of humanism. But let there be no misunderstanding: it is not that a real man, the object of knowledge, philosophical reflection, or technical intervention, has been substituted for the illusion of the theologians. The man described for us, whom we are invited to free, is already in himself the effect of a subjection much more profound than himself. A 'soul' inhabits him and brings him into existence, which is itself a factor in the mastery that power exercises over the body. The soul is the effect and instrument of a political anatomy; the soul is the prison of the body""(Foucault, Discipline and Punish pg. 29-30)

"In discipline, it is the subjects[think "I" for Hofstadter] who have to be seen. Their visibility assures the hold of the power that is exercised over them. It is the fact of being constantly seen, of being able always to be seen, that maintains the disciplined individual in his subjection"(Foucault, Discipline and Punish pg. 187)

And some more:

"The individual is no doubt the fictitious atom of an 'ideological' representation of society; but he is also a reality fabricated by this specific technology of power I have called 'discipline'(Foucault, pg. 194)

In Foucault, self-consciousness is the the result of a technique of power being exercised on us by those in power.

"Since we perceive not particles interacting but macroscopic patterns in which certain thing s push other things around with a blurry causality, and since the Grand Pusher in and of our bodies is our "I", and since our bodies push the rest of the world around, we are left with no choice but to conclude that the "I" is where the causality buck stops."(Hofstadter, pg. 217 Nookbook).

This "I" gains structure as we get older:

"We begin life with the most elementary sorts of feedback about ourselves, which stimulate us to formulate categories for our most obvious body parts, and building on this basic pedestal, we soon develop a sense of our bodies as flexible physical objects. In the meantime, as we receive rewards for various actions and punishments for others, we being to develop a more abstract sense of "good" and "bad", as well as notions of guilt and pride, and our sense of ourselves as abstract entities that have the power to decide to make things happen ... begins to take root."(Hofstadter, pg. 218, Nookbook)

This remarkable passage bears a striking resemblance to some ideas of Michel Foucault in

*Discipline and Punish,*one of the most important books of the 20th century, and one of the most influential on me:"This real, noncorporal soul, is not a substance; it is the element in which are articulated the effects of a certain type of power and the reference of a certain type of knowledge, the machinery by which the power relations give rise to a possible corpus of knowledge, and knowledge extends and reinforces the effects of this power. On this reality-reference, various concepts have been constructed and domains of analysis carried out: psyche, subjectivity, personality, consciousness etc.; on it have been built scientific techniques and discourses, and the moral claims of humanism. But let there be no misunderstanding: it is not that a real man, the object of knowledge, philosophical reflection, or technical intervention, has been substituted for the illusion of the theologians. The man described for us, whom we are invited to free, is already in himself the effect of a subjection much more profound than himself. A 'soul' inhabits him and brings him into existence, which is itself a factor in the mastery that power exercises over the body. The soul is the effect and instrument of a political anatomy; the soul is the prison of the body""(Foucault, Discipline and Punish pg. 29-30)

"In discipline, it is the subjects[think "I" for Hofstadter] who have to be seen. Their visibility assures the hold of the power that is exercised over them. It is the fact of being constantly seen, of being able always to be seen, that maintains the disciplined individual in his subjection"(Foucault, Discipline and Punish pg. 187)

And some more:

"The individual is no doubt the fictitious atom of an 'ideological' representation of society; but he is also a reality fabricated by this specific technology of power I have called 'discipline'(Foucault, pg. 194)

In Foucault, self-consciousness is the the result of a technique of power being exercised on us by those in power.

## Sunday, June 19, 2011

### 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

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

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

### Incompleteness Theorem -- part 5, the last one, I promise

OK, so I've gone through the demonstration that there are undecidable propositions inside any formal system sufficient for addition, multiplication, and the basic logical operations of the elementary theory of whole numbers. From here it's actually pretty easy to see why the consistency of the formal system, which I'm referring to as PM, cannot be proven within the system. Again, I will stay as close as I can, up to the ability of this editor to represent it, to Goedel's notation.

Let Wid(c) be defined as the statement that there exists a formula x such that it cannot be derived from the set of formulae c. That is, (Ex)[Form(x) & ~(Bew(x))]. Here Bew means provable from the formulae c. Literally, There exists an x such that x is a formula of PM and x is not provable from the formulae c.

Now we have to prove this. Remember that 17 Gen r is not provable within PM. So, Wid(c)-->~(Bew(17 Gen r)). That is, as long as there are unprovable propositions, the undecidable one from before is unprovable. But the fact that for any x, 17 Gen r is not provable is equivalent to the statement(for all x)q(Z(x),Z(p)), recalling that 17 Gen r = p(Z(p)). Therefore, if PM is consistent, we can prove 17 Gen r! This is a contradiction of Wid(c)-->~(Bew(17 Gen r)) above. Here we use the fact that 17 Gen r states its own unprovability, so to prove it unprovable is to prove it.

Thus, a formal system like PM cannot prove its own consistency. Well, what's so big about that? For one thing, suppose you have a bigger system that contains PM, can it prove it's own consistency? It seems it can't because the argument above would apply to it and we would have an infinite regress. Please let me know if I'm wrong about this.

Next time I will return to

Let Wid(c) be defined as the statement that there exists a formula x such that it cannot be derived from the set of formulae c. That is, (Ex)[Form(x) & ~(Bew(x))]. Here Bew means provable from the formulae c. Literally, There exists an x such that x is a formula of PM and x is not provable from the formulae c.

Now we have to prove this. Remember that 17 Gen r is not provable within PM. So, Wid(c)-->~(Bew(17 Gen r)). That is, as long as there are unprovable propositions, the undecidable one from before is unprovable. But the fact that for any x, 17 Gen r is not provable is equivalent to the statement(for all x)q(Z(x),Z(p)), recalling that 17 Gen r = p(Z(p)). Therefore, if PM is consistent, we can prove 17 Gen r! This is a contradiction of Wid(c)-->~(Bew(17 Gen r)) above. Here we use the fact that 17 Gen r states its own unprovability, so to prove it unprovable is to prove it.

Thus, a formal system like PM cannot prove its own consistency. Well, what's so big about that? For one thing, suppose you have a bigger system that contains PM, can it prove it's own consistency? It seems it can't because the argument above would apply to it and we would have an infinite regress. Please let me know if I'm wrong about this.

Next time I will return to

*I am a Strange Loop.*## Saturday, June 18, 2011

### Incompleteness Theorem -- Part 4

Goedel ends his 46 definitions with the definition for provable:

46. Bew(x):=(Ey)yBx

This definition refers to definition 45 and says: " There exists a y such that y is a proof of x."

There is a theorem, provable by induction, that I will take as a given.

Theorem 0: For every recursive relation R(x1,...,xn) there exists an n place relation sign r between the free variables u1,...,un such that for all n-tuples of numbers (x1,...,xn):

R(x1,...,xn) -->Bew[Sb(r (u1,...,un)=(Z(x1),...,Z(xn))]

_

R(x1,...,xn)-->Bew[Neg[Sb(r u1,...,un)=(Z(x1),...,Z(xn))]]

What this says is that if the relation R holds between the n-tuple of numbers x1,...,xn, then there exists a relation r between variables of PM such that the proposition obtained by substituting the numerals Z(x1),...,Z(xn) corresponding to the n-tuple of numbers into the n variables of r, the result is provable. Note: numerals are repeated applications of the successor function to zero to obtain an expression for a number, say xi, inside the formal system I'm calling PM. This completes the translation from the numerical relation R to the relation r between signs in PM. If the negation of R is true, then the provable proposition is the negation of the proposition you get by performing the substitution into r.

Basically, if a relationship R holds between the natural numbers x1,..,xn, then in PM there is a corresponding relation between the signs(numerals) for the numbers. If the relation does not hold between the numbers, then the corresponding relation between the numerals does not hold.

Now that we have this we can proceed to the statement of the First Incompleteness Theorem:

Definition: A formal system like PM satisfies omega-consistency if there is no formula a for which a(Z(n)) is true for each n and at the same time (for all n)a(Z(n)) is false.

Theorem 1: For every omega-consistent recursive class K of formulas[like the formulas of PM] there are recursive class signs r such that neither v Gen r nor Neg(v Gen r) belongs to FLG(K).

Recall v Gen R:=for all v, R is true.

Also, FLG(K) is the set of consequences of K.

Now, define the relation Q(x,y) between the natural numbers x and y as the relation that exists when the formula corresponding to the Goedel number x is not a proof of the formula obtained when Z(y) is substituted in for the free variable(whose Goedel number sequence is 19) of the formula whose Goedel number is y. That is, and this is important, the numeral for the Goedel number of y is substituted into the formula y's own free variable.

By Theorem 0, there is a relation q in PM such that q(Z(x),Z(y)) is provable in PM whenever Q(x,y) is true, and the negation of q(Z(x),Z(y)) is provable whenever Q(x,y) is false. The relation q is exactly the relation that obtains between Z(x) and Z(y) when the formula x is not a proof of y(Z(y)).

Goedel introduces some abbreviations: p=17 Gen q and r=Sb(q 19=Z(p)).

Now we consider the proposition 17 Gen r=17 Gen(Sb(q 19=Z(p))=Sb(p 19=Z(p))=p(Z(p).

Now, this abbreviation extravaganza actually obscures what is happening here. The wikipedia page has it right: 17 Gen r = (for all y)q(y,Z(p)) . That is, for all y, we have y is not a proof of p(Zp), which is the same as saying p(Zp) is not provable. Now, and this is the interesting part, p(Z(p))

Assumption 1: Assume (for all y)q(y,Z(p)) is provable, then there is a formula n such that n is a proof of (for all y)q(y,Z(p)). Substituting n for y we get that q(n,Z(P)) is true.

Now, p(Z(p))

Assumption 2: Assume ~(for all y)q(y,Z(p)). By Theorem 0 we have q(z(n),Z(p)) for every n. But our assumption is ~(for all y)q(y,Z(p)), which contradicts omega-consistency.

46. Bew(x):=(Ey)yBx

This definition refers to definition 45 and says: " There exists a y such that y is a proof of x."

There is a theorem, provable by induction, that I will take as a given.

Theorem 0: For every recursive relation R(x1,...,xn) there exists an n place relation sign r between the free variables u1,...,un such that for all n-tuples of numbers (x1,...,xn):

R(x1,...,xn) -->Bew[Sb(r (u1,...,un)=(Z(x1),...,Z(xn))]

_

R(x1,...,xn)-->Bew[Neg[Sb(r u1,...,un)=(Z(x1),...,Z(xn))]]

What this says is that if the relation R holds between the n-tuple of numbers x1,...,xn, then there exists a relation r between variables of PM such that the proposition obtained by substituting the numerals Z(x1),...,Z(xn) corresponding to the n-tuple of numbers into the n variables of r, the result is provable. Note: numerals are repeated applications of the successor function to zero to obtain an expression for a number, say xi, inside the formal system I'm calling PM. This completes the translation from the numerical relation R to the relation r between signs in PM. If the negation of R is true, then the provable proposition is the negation of the proposition you get by performing the substitution into r.

Basically, if a relationship R holds between the natural numbers x1,..,xn, then in PM there is a corresponding relation between the signs(numerals) for the numbers. If the relation does not hold between the numbers, then the corresponding relation between the numerals does not hold.

Now that we have this we can proceed to the statement of the First Incompleteness Theorem:

Definition: A formal system like PM satisfies omega-consistency if there is no formula a for which a(Z(n)) is true for each n and at the same time (for all n)a(Z(n)) is false.

Theorem 1: For every omega-consistent recursive class K of formulas[like the formulas of PM] there are recursive class signs r such that neither v Gen r nor Neg(v Gen r) belongs to FLG(K).

Recall v Gen R:=for all v, R is true.

Also, FLG(K) is the set of consequences of K.

Now, define the relation Q(x,y) between the natural numbers x and y as the relation that exists when the formula corresponding to the Goedel number x is not a proof of the formula obtained when Z(y) is substituted in for the free variable(whose Goedel number sequence is 19) of the formula whose Goedel number is y. That is, and this is important, the numeral for the Goedel number of y is substituted into the formula y's own free variable.

By Theorem 0, there is a relation q in PM such that q(Z(x),Z(y)) is provable in PM whenever Q(x,y) is true, and the negation of q(Z(x),Z(y)) is provable whenever Q(x,y) is false. The relation q is exactly the relation that obtains between Z(x) and Z(y) when the formula x is not a proof of y(Z(y)).

Goedel introduces some abbreviations: p=17 Gen q and r=Sb(q 19=Z(p)).

Now we consider the proposition 17 Gen r=17 Gen(Sb(q 19=Z(p))=Sb(p 19=Z(p))=p(Z(p).

Now, this abbreviation extravaganza actually obscures what is happening here. The wikipedia page has it right: 17 Gen r = (for all y)q(y,Z(p)) . That is, for all y, we have y is not a proof of p(Zp), which is the same as saying p(Zp) is not provable. Now, and this is the interesting part, p(Z(p))

*IS*17 Gen r!Assumption 1: Assume (for all y)q(y,Z(p)) is provable, then there is a formula n such that n is a proof of (for all y)q(y,Z(p)). Substituting n for y we get that q(n,Z(P)) is true.

Now, p(Z(p))

*IS*17 Gen r so by Theorem 0, given that n proves p(Z(p)) we know that Q(n,p(Z(p)) is false, we have ~q(n,Z(p)). This contradicts q(n,Z(P)) shown above.Assumption 2: Assume ~(for all y)q(y,Z(p)). By Theorem 0 we have q(z(n),Z(p)) for every n. But our assumption is ~(for all y)q(y,Z(p)), which contradicts omega-consistency.

## Friday, June 17, 2011

### Incompleteness Theorem -- part 3

See the great wikipedia page on the proof of Goedel's incompleteness theorem:

http://en.wikipedia.org/wiki/Proof_sketch_for_G%C3%B6del%27s_first_incompleteness_theorem

Now I'll go through an examination of Goedel numbering and pick some hilights from the 45 recursive definitions Goedel gives before moving on to the main proof in the next post.

We assign numbers to the elementary signs as follows:

"0" ... 1

"f" ... 3

"~" ... 5

"or" ... 7

"For all" ... 9

"(" ... 11

")" ... 13

and variables of type n are assigned values of the form p(to the power n) where p is a prime number. So, a type-I variable, a variable standing for a number, is just a prime to the first power. So, 17, 19, 23, etc.. are type one variables.

To move from a number sequence to a Goedel number, we just take the primes in increasing order and raise them to the powers of the signs. For example, 11,17,13 becomes 2(to the 11 power) times 3(to the 17 power) times 5(to the 13 power). Here, 17 is a variable, say x, so 11,17,13 is the sequence for "(x)".

I'm going to try and stay close to Goedel's proof despite the fact that sometimes the notation is a little difficult to understand. I'm going to go through some of the hilights of the 45 definitions he gives before provability. I will then borrow some discussion from the wikipedia page above regarding Goedel numbers of proofs.

Ex="There exists an x". ":=" means "defined to be"

So, here we go.

1.x|y:=(Ez)[z<=x & x=y.z]

x is divisible by y.

___

2. Prim(x):=(Ez)[z<=x & z not = 1 & z not=x &x|z] & x>1.

This reads: There does not exist a z such that z is less than or equal to z with z not =1, z not=x and where x is divisible by z, and also x is greater than 1. A simpler way of saying this is that the only numbers that divide x are 1 and x, and x >1. In other words, x is prime. Notice that 2. used x|y, which is 1.

Skipping to number 8

8.x*y:=(the smallest z){z <=Pr(l(x)+l(y))](to the x+y power) & (for all n)[n<=l(x)-->nGlz=nGlx] & (for all n)[0 < n <= l(y)-->(n+l(x))Glz=nGly]}

This expression uses l(x) from 7., nGLx from 6., and nPrx from 5. So, this is a function recursively defined in terms of previous functions. I know it looks long, but if you write it out it's horrible:

x*y is defined to be the smallest z such that z is smalller than the prime number at place length of the number sequence assigned to x plus length of the number sequence assigned to y to the x=y power and for all n, if n is less than or equal to the length of the number sequence of x then the nth term in the number sequence assigned to z equals the nth term of the number sequence assigned to x and for all n, if n is bigger than zero and less than or equal to the length of the number sequence assigned to y, then the n + length of the number sequence assigned to x term in the number series assigned to z equals the nth term in the number series assigned to y.

All this definition say is that you take two number series and stick the one for y to the right of the one for x. This corresponds in PM to putting two series of signs, for x and y, next to each other. What this illustrates is that the recursive way Goedel does these definitions allows us to avoid having to write out gazillions of symbols and words. By the time you get to 45, these are some pretty big expressions.

Armed with 8 we can start with

9. R(x):=2(to the x power)

and get

10.E(x):=R(11)*x*R(13) to mean putting the Goedel number for "(" with "x" and ")" to yield "(x)" -- 11 is the exponent assigned to "(" and 13 goes to ")". Imagine if this had to be written out!

We can also get

13. Neg(x):=R(5)*E(x) to mean the negation of x or "~(x)"

14. xDisy:=E(x)*R(7)*E(y) means (x) or (y).

15.xGeny:=R(x)*R(9)*E(y) means: "y is true for all x".

Here's part of the definition of substituting y for the nth term of x(number 27)

start with x=u*R(nGlx)*v and substitute y in to yield a number given by u*y*v, where n=l(u)+1(which means that y was subtituted in the place in the number immediately following u, as we wanted).

Later on, number 31, we get the definition of substituting in for a free variable.

Goedel goes on to define axioms, formulas, immediate consequence, and proof.

We end with provable, number 46.

To think about provability we consider a deduction rule D as a relation between the first n-1 formulas of a list and the nth formula. The Goedel number for the first n-1 will stand in numerical relation R to the Goedel number for the nth formula exactly when the first n-1 formulas imply the nth formula by the rule D. So, if we have a list of k deduction rules, then the n-1 formulas is a proof of the nth if the Goedel number of the first n-1 formulas stands in at least 1 of the k corresponding numerical relations to the nth Goedel number.

http://en.wikipedia.org/wiki/Proof_sketch_for_G%C3%B6del%27s_first_incompleteness_theorem

Now I'll go through an examination of Goedel numbering and pick some hilights from the 45 recursive definitions Goedel gives before moving on to the main proof in the next post.

We assign numbers to the elementary signs as follows:

"0" ... 1

"f" ... 3

"~" ... 5

"or" ... 7

"For all" ... 9

"(" ... 11

")" ... 13

and variables of type n are assigned values of the form p(to the power n) where p is a prime number. So, a type-I variable, a variable standing for a number, is just a prime to the first power. So, 17, 19, 23, etc.. are type one variables.

To move from a number sequence to a Goedel number, we just take the primes in increasing order and raise them to the powers of the signs. For example, 11,17,13 becomes 2(to the 11 power) times 3(to the 17 power) times 5(to the 13 power). Here, 17 is a variable, say x, so 11,17,13 is the sequence for "(x)".

I'm going to try and stay close to Goedel's proof despite the fact that sometimes the notation is a little difficult to understand. I'm going to go through some of the hilights of the 45 definitions he gives before provability. I will then borrow some discussion from the wikipedia page above regarding Goedel numbers of proofs.

Ex="There exists an x". ":=" means "defined to be"

So, here we go.

1.x|y:=(Ez)[z<=x & x=y.z]

x is divisible by y.

___

2. Prim(x):=(Ez)[z<=x & z not = 1 & z not=x &x|z] & x>1.

This reads: There does not exist a z such that z is less than or equal to z with z not =1, z not=x and where x is divisible by z, and also x is greater than 1. A simpler way of saying this is that the only numbers that divide x are 1 and x, and x >1. In other words, x is prime. Notice that 2. used x|y, which is 1.

Skipping to number 8

8.x*y:=(the smallest z){z <=Pr(l(x)+l(y))](to the x+y power) & (for all n)[n<=l(x)-->nGlz=nGlx] & (for all n)[0 < n <= l(y)-->(n+l(x))Glz=nGly]}

This expression uses l(x) from 7., nGLx from 6., and nPrx from 5. So, this is a function recursively defined in terms of previous functions. I know it looks long, but if you write it out it's horrible:

x*y is defined to be the smallest z such that z is smalller than the prime number at place length of the number sequence assigned to x plus length of the number sequence assigned to y to the x=y power and for all n, if n is less than or equal to the length of the number sequence of x then the nth term in the number sequence assigned to z equals the nth term of the number sequence assigned to x and for all n, if n is bigger than zero and less than or equal to the length of the number sequence assigned to y, then the n + length of the number sequence assigned to x term in the number series assigned to z equals the nth term in the number series assigned to y.

All this definition say is that you take two number series and stick the one for y to the right of the one for x. This corresponds in PM to putting two series of signs, for x and y, next to each other. What this illustrates is that the recursive way Goedel does these definitions allows us to avoid having to write out gazillions of symbols and words. By the time you get to 45, these are some pretty big expressions.

Armed with 8 we can start with

9. R(x):=2(to the x power)

and get

10.E(x):=R(11)*x*R(13) to mean putting the Goedel number for "(" with "x" and ")" to yield "(x)" -- 11 is the exponent assigned to "(" and 13 goes to ")". Imagine if this had to be written out!

We can also get

13. Neg(x):=R(5)*E(x) to mean the negation of x or "~(x)"

14. xDisy:=E(x)*R(7)*E(y) means (x) or (y).

15.xGeny:=R(x)*R(9)*E(y) means: "y is true for all x".

Here's part of the definition of substituting y for the nth term of x(number 27)

start with x=u*R(nGlx)*v and substitute y in to yield a number given by u*y*v, where n=l(u)+1(which means that y was subtituted in the place in the number immediately following u, as we wanted).

Later on, number 31, we get the definition of substituting in for a free variable.

Goedel goes on to define axioms, formulas, immediate consequence, and proof.

We end with provable, number 46.

To think about provability we consider a deduction rule D as a relation between the first n-1 formulas of a list and the nth formula. The Goedel number for the first n-1 will stand in numerical relation R to the Goedel number for the nth formula exactly when the first n-1 formulas imply the nth formula by the rule D. So, if we have a list of k deduction rules, then the n-1 formulas is a proof of the nth if the Goedel number of the first n-1 formulas stands in at least 1 of the k corresponding numerical relations to the nth Goedel number.

## Thursday, June 16, 2011

### Incompleteness Theorem -- part 2

See the following by Torkel Franzen for insight into the incompleteness theorems:

http://www.hashref.com/summaries/GodelsTheorem.pdf

Important for Hofstadter is the notion of recursion. This notion also plays a central role in the Goedel's proof. Goedel gives the following definition:

phi(x1,x2,...,xn) is said to be

phi(0,x2,...,xn)=psi(x2,...,xn) and phi(k+1,x2,...,xn)=mu(k,phi(k,x2,...,xn),x2,...,xn).

So, phi is initialized by psi, and increasing values of x1 are given in terms of mu where he have substituted the previous k's phi value in for the value before x2 in mu. Of course we could iterated back from k+1 to zero by back substituting. Oddly, the mu is a function of more variables than phi itself and yet phi is written recursively in terms of it. Well, I guess I don't understand that too well. But then Goedel adds the clause "or results from any of the preceding functions by substitution"; he then gives a footnote with a nice example:

phi_k(x1,x2)=phi_p[phi_q(x1,x2),phi_r(x2)] (p,q,r < k)

This is more like what I think of when I think of recursion, and it allows me to think of a function as recursively defined in term of the previous two functions. The mu thing hasn't really caused me any trouble understanding the rest of the proof, but if anyone out there wants to clear up the mu business that would be cool. When he defines a recursive relation R between natural numbers as being defined by the zeroes of a recursive function phi; that is, R(x1,...,xn) is given by the points x1,...,xn that satisfy phi(x1,...,xn)=0, where phi is recursively defined.

He then states some theorems that make good sense:

1. A function(relation) obtained from recursive functions is recursive.

2. If R and S are recursive, then so are negation R and (R or S) and (R and S).

3. If the functions phi(x) and psi(y) are recursive, so is the relation phi(x)=psi(y).

4. If phi(x) and R(y,z) are recursive then so are the relations

S(x,z)=(there exists a y such that)[y <= phi(x) and R(y,z) holds]

T(x,z)=(for all y)[if y <=phi(x) then R(y,z) holds]

The recursive nature of these definitions plays an important role in the building up of 45 the definitions that he needs to show that unprovability is expressible in terms of PM:

"The functions x+y, x.y, and x(to the y power), as well as the relations x < y and x=y, are recursive, as we readily see. Starting from these notions, we now define a number of functions(relations) 1-45, each of which is defined in terms preceding ones by the procedures given in Theorems 1-4[above] ... Each of the functions (relations) 1-45, among them occur, for example, the notions "FORMULA, "AXIOM", and "IMMEDIATE CONSEQUENCE", is recursive."[van Heijenoort,

I take for granted that he succeeded in showing that these things can be expressed in PM. From there the proof of the incompleteness theorem is not so bad.

Hofstadter mentions the recursive formulas when he talks about Goedel and the Fibonacci sequence, which is a sequence of numbers such that each number is the sum of the previous two(starting with the third number, of course): 1,1,2,3,5,8,13,21,...

"...the axioms of PM would play the role of Fibonacci seeds 1 and 2, and the rules of inference of PM would play the role of adding the two most recent numbers."(Hofstadter pp. 160-161 Nookbook)

I will go into more detail regarding the first 45 recursive definitions in the next post.

http://www.hashref.com/summaries/GodelsTheorem.pdf

Important for Hofstadter is the notion of recursion. This notion also plays a central role in the Goedel's proof. Goedel gives the following definition:

phi(x1,x2,...,xn) is said to be

*recursively defined*in terms of psi(x1,x2,,...,x(n-1)) and mu(x1,x2,...,x(n+1)) ifphi(0,x2,...,xn)=psi(x2,...,xn) and phi(k+1,x2,...,xn)=mu(k,phi(k,x2,...,xn),x2,...,xn).

So, phi is initialized by psi, and increasing values of x1 are given in terms of mu where he have substituted the previous k's phi value in for the value before x2 in mu. Of course we could iterated back from k+1 to zero by back substituting. Oddly, the mu is a function of more variables than phi itself and yet phi is written recursively in terms of it. Well, I guess I don't understand that too well. But then Goedel adds the clause "or results from any of the preceding functions by substitution"; he then gives a footnote with a nice example:

phi_k(x1,x2)=phi_p[phi_q(x1,x2),phi_r(x2)] (p,q,r < k)

This is more like what I think of when I think of recursion, and it allows me to think of a function as recursively defined in term of the previous two functions. The mu thing hasn't really caused me any trouble understanding the rest of the proof, but if anyone out there wants to clear up the mu business that would be cool. When he defines a recursive relation R between natural numbers as being defined by the zeroes of a recursive function phi; that is, R(x1,...,xn) is given by the points x1,...,xn that satisfy phi(x1,...,xn)=0, where phi is recursively defined.

He then states some theorems that make good sense:

1. A function(relation) obtained from recursive functions is recursive.

2. If R and S are recursive, then so are negation R and (R or S) and (R and S).

3. If the functions phi(x) and psi(y) are recursive, so is the relation phi(x)=psi(y).

4. If phi(x) and R(y,z) are recursive then so are the relations

S(x,z)=(there exists a y such that)[y <= phi(x) and R(y,z) holds]

T(x,z)=(for all y)[if y <=phi(x) then R(y,z) holds]

The recursive nature of these definitions plays an important role in the building up of 45 the definitions that he needs to show that unprovability is expressible in terms of PM:

"The functions x+y, x.y, and x(to the y power), as well as the relations x < y and x=y, are recursive, as we readily see. Starting from these notions, we now define a number of functions(relations) 1-45, each of which is defined in terms preceding ones by the procedures given in Theorems 1-4[above] ... Each of the functions (relations) 1-45, among them occur, for example, the notions "FORMULA, "AXIOM", and "IMMEDIATE CONSEQUENCE", is recursive."[van Heijenoort,

*From Frege to Goedel*, pg. 603 -- this book is a compendium of classic papers in mathematical logic including Goedel's incompleteness theorems.]I take for granted that he succeeded in showing that these things can be expressed in PM. From there the proof of the incompleteness theorem is not so bad.

Hofstadter mentions the recursive formulas when he talks about Goedel and the Fibonacci sequence, which is a sequence of numbers such that each number is the sum of the previous two(starting with the third number, of course): 1,1,2,3,5,8,13,21,...

"...the axioms of PM would play the role of Fibonacci seeds 1 and 2, and the rules of inference of PM would play the role of adding the two most recent numbers."(Hofstadter pp. 160-161 Nookbook)

I will go into more detail regarding the first 45 recursive definitions in the next post.

## Tuesday, June 14, 2011

### Incompleteness Theorem -- part 1

"The most comprehensive formal systems that have been set up hitherto are the system of

It was with these words that Goedel announced what he was going to demonstrate. As if by magic, Goedel maps statements within formal system to statemens ABOUT statements within formal systems. Thus, Goedel demonstrates that certain true propositions in the language map to true assertions that they are unprovable -- assuming, of course, the formal system is consistent. Goedel begins by giving a sketch of the main idea of the proof. In the below, we will use

The basic signs of PM are mapped onto numbers in an ingenious way, so that proposiions, like, "2+2=4", are mapped to a number. A more complicated expression, like "If x and y are bigger than z and w, then x+y is bigger than z+w" is also mapped to a number, and is mapped in such a way that the number, as well as it decomposition into factors, expresses important facts about the proposition, most relevantly for our purposes the fact that the proposition is provable within PM. What Goedel demonstrates is that there are some propositions that map to numbers in such a way that these numerical relationships reveal the unprovability of the proposition within PM.

Goedel expresses this in his sketch of the main proof by defining a class K of natural numbers defined in such a way that the natural number

Here we can see the sort of thing Hofstadter is going to make a big deal of: from statements about numbers in PM we are able to get layers of meaning about statements, including proofs that statements themselves are unprovable. These "meta" statements are seen to emerge out of the

In future posts I will work carefully through Goedel's proof of undecidable propositions and also the fact that these systems cannot prove their own consistency.

*Principia Mathematica(PM*) on the one hand and the Zermelo-Fraenkel axiom system of set theory (further developed by J. von Neumann) on the other. These two systems are so comprehensive that in them all methods of proof today used in mathematics are formalized, that is, reduced to a few axioms and rules of inference. One might therefore conjecture that these axioms and rules of inference are sufficient to decide*any*mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case ...This situation is not in any way due to the special nature of the systems that have been set up but holds for a wide class of formal systems..."*On Formally Undecidable Propositions of Principia Mathematica and Related Systems I**--*Kurt GoedelIt was with these words that Goedel announced what he was going to demonstrate. As if by magic, Goedel maps statements within formal system to statemens ABOUT statements within formal systems. Thus, Goedel demonstrates that certain true propositions in the language map to true assertions that they are unprovable -- assuming, of course, the formal system is consistent. Goedel begins by giving a sketch of the main idea of the proof. In the below, we will use

*Principia Mathematica (PM)*as the formal system.The basic signs of PM are mapped onto numbers in an ingenious way, so that proposiions, like, "2+2=4", are mapped to a number. A more complicated expression, like "If x and y are bigger than z and w, then x+y is bigger than z+w" is also mapped to a number, and is mapped in such a way that the number, as well as it decomposition into factors, expresses important facts about the proposition, most relevantly for our purposes the fact that the proposition is provable within PM. What Goedel demonstrates is that there are some propositions that map to numbers in such a way that these numerical relationships reveal the unprovability of the proposition within PM.

Goedel expresses this in his sketch of the main proof by defining a class K of natural numbers defined in such a way that the natural number

*n*is in the class K if and only if the statement identified by that number is unprovable. The way we get an unprovable proposition is to define a proposition S, interpreted in PM to mean that its corresponding number, call it*q*, belongs to the class K. Now, suppose S is provable, then it would be true, which would mean that*q*belongs to K, which means that S is unprovable. On the other hand, if the negation of S is provable, then*q*does not belong to K, in which case S is provable. So, if S is provable it is unprovable, a contradiction; if negation S is provable, then S is provable, a contradiction. Thus, S is undecidable.Here we can see the sort of thing Hofstadter is going to make a big deal of: from statements about numbers in PM we are able to get layers of meaning about statements, including proofs that statements themselves are unprovable. These "meta" statements are seen to emerge out of the

*strange loop*structure.In future posts I will work carefully through Goedel's proof of undecidable propositions and also the fact that these systems cannot prove their own consistency.

## Sunday, June 12, 2011

### I Am a Strange Loop -- Commentary on Chapter 8

Chapter 8 is where get the definition of the

He gives the example of Escher's

As noted before, the self, despite its epiphenomenal status, is a prime mover, a real causal agent in the world; that is, it is real:

"...the thesis of this book is that we ourselves -- not our bodies, but our

So, how do we get from a drawn to a drawer? How do we move from one level to another. Well, here's a shot at it. The self reflection we get in consciousness, when we turn the camera on ourselves, creates a mental structure that is present exactly when that reflection occurs. It persists as long as there is self-reflection. When self-reflection ceases, the self ceases. But we are conditioned by experience to think of ourselves as acting. Thus the self is one of those abstract notions like those Hofstadter mentions earlier in the book that is causally effective in its own right. The interesting thing here, is that this particular structure is the result of "loopy" behavior. The loop brings this entity into existence. It is paradoxical, difficult to follow, but there it is.

I think I will digress a little on the notion of the self and on self-reflection. What are some of the things that accompany self reflection? If, as Fichte says, our self-reflection occurs in a social, and striving, context, then the structures in the self have to do with these struggles. Hegel builds ideas of Master and Slave consciousness out of his philosophy of history. There can be consciousness associated with oppressor and oppressed, for example. Perhaps these yield different views of the world because fundamental self-consciousness is different. Is there such a thing as "pure" self-consciousness? I wonder if the instructions of some meditation tapes I have, "I am that which is having the experiences", gives us a perspective that frees us from received self structures and yields an unmodified self, separate from everything. Well, probably not, but it's a thought.

Starting with my next entry I will begin a series on Goedel's Incompleteness Theorems based on my reading of the original paper and Braithewaite's introduction. This will tie into chapter 9.

*strange loop*, the "I" being itself one of these. Hofstadter defines a*strange loop*as "not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive "upwards" shifts turn out to give rise to a closed cycle."He gives the example of Escher's

*Drawing Hands,*a picture of a left hand and right hand drawing each other: "Here, the abstract shift in level would be the upward leap from*drawn*to*drawer";*believe it or not there is a wikipedia page on*Drawing Hands*--(http*://en.wikipedia.org/wiki/Drawing_Hands*) -- check it out!As noted before, the self, despite its epiphenomenal status, is a prime mover, a real causal agent in the world; that is, it is real:

"...the thesis of this book is that we ourselves -- not our bodies, but our

*selves --*are strange loops, and so if all strange loops were illusions, then we would all be illusions, and that would be a great shame. So it's fortunate that some strange loops exist in the real world."(Hofstadter, pg. 134 Nookbook)So, how do we get from a drawn to a drawer? How do we move from one level to another. Well, here's a shot at it. The self reflection we get in consciousness, when we turn the camera on ourselves, creates a mental structure that is present exactly when that reflection occurs. It persists as long as there is self-reflection. When self-reflection ceases, the self ceases. But we are conditioned by experience to think of ourselves as acting. Thus the self is one of those abstract notions like those Hofstadter mentions earlier in the book that is causally effective in its own right. The interesting thing here, is that this particular structure is the result of "loopy" behavior. The loop brings this entity into existence. It is paradoxical, difficult to follow, but there it is.

I think I will digress a little on the notion of the self and on self-reflection. What are some of the things that accompany self reflection? If, as Fichte says, our self-reflection occurs in a social, and striving, context, then the structures in the self have to do with these struggles. Hegel builds ideas of Master and Slave consciousness out of his philosophy of history. There can be consciousness associated with oppressor and oppressed, for example. Perhaps these yield different views of the world because fundamental self-consciousness is different. Is there such a thing as "pure" self-consciousness? I wonder if the instructions of some meditation tapes I have, "I am that which is having the experiences", gives us a perspective that frees us from received self structures and yields an unmodified self, separate from everything. Well, probably not, but it's a thought.

Starting with my next entry I will begin a series on Goedel's Incompleteness Theorems based on my reading of the original paper and Braithewaite's introduction. This will tie into chapter 9.

### I Am a Strange Loop -- Commentary on Chapter 7

In Chapter 7, Hofstadter embarks on some of his central notions, most central of all is the concept of the "I". The "I" is a concept that has obsessed philosophers for centuries, especially it seems, the idealists, who believed that fundamental to reality were conscious perceptions, the physical world being merely a supposition of the self. The "I" is a transcendental point, a center of perception that itself is not part of empirical world. The analytic philosopher, Wittgenstein, has a famous drawing of the self not being in the field of perception in his

For Hofstadter, the self is an epiphenomena, by that he means a phenomena which is the result of many things occuring at more micro levels, having no reality itself. He relates a story of reaching into a box of envelopes without looking and feeling what he interprets as a marble in the middle of the box. He looks into the box and doesn't see a marble.

"But then, as soon as I grasped the whole set of envelopes as before, there it was again, as solid as ever!"(Hofstadter, pg. 124 Nookbook)

He concluded that there wasn't a marble -- but what was he experiencing?

"It was an

The self is an epiphenomenon like the marble:

"The thesis of this book is that in a nonembryonic, noninfantile human brain, there is a special type of abstract structure or pattern that plays the same role as does that precise alignment of layers of paper and glue -- an abstract pattern that gives rise to what

Hofstadter says that it is the "collection of desires and beliefs"(pg. 128) that sets behavior in motion; it is "this "I" that is the prime mover", not elementary particles or molecules.

All of this makes a lot of sense to me: the behavior of the zillions of elementary particles add up statistically to conceptual units that are more properly seen as the basis of behavior than the "really existing" particles themselves. This is a very nice move on Hofstadter's part: he shows that the "macro", i.e., "I" level explanation of behavior is more to the point than a reductionist one. The metaphors and analogies he gives early in the book pay off very well here. It is worth reading his notion of "Thinkodynamics" in chapter 2 again.

In Chapter 8, Hofstadter finally defines the

*Tractatus Logico Philosophicus --*you should check this book out if you haven't read it. The "I", even for idealists is not empirically real, but rather is part of a transcendental reality.For Hofstadter, the self is an epiphenomena, by that he means a phenomena which is the result of many things occuring at more micro levels, having no reality itself. He relates a story of reaching into a box of envelopes without looking and feeling what he interprets as a marble in the middle of the box. He looks into the box and doesn't see a marble.

"But then, as soon as I grasped the whole set of envelopes as before, there it was again, as solid as ever!"(Hofstadter, pg. 124 Nookbook)

He concluded that there wasn't a marble -- but what was he experiencing?

"It was an

*epiphenomenon*caused by the fact that, for each envelope, at the central vertex of the 'V' made by its flap, there is a triple layer of paper as well as a thin layer of glue. An unintended consequence of this innocent design decision is that when you squeeze down on a hundred such envelopes all precisely aligned with each other, you can't compress that little zone as much as the other zones -- it resists compression." (Hofstadter, pg. 124, Nookbook)The self is an epiphenomenon like the marble:

"The thesis of this book is that in a nonembryonic, noninfantile human brain, there is a special type of abstract structure or pattern that plays the same role as does that precise alignment of layers of paper and glue -- an abstract pattern that gives rise to what

*feels*like a self." (Hofstadter, pp. 126-127 Nookbook)Hofstadter says that it is the "collection of desires and beliefs"(pg. 128) that sets behavior in motion; it is "this "I" that is the prime mover", not elementary particles or molecules.

All of this makes a lot of sense to me: the behavior of the zillions of elementary particles add up statistically to conceptual units that are more properly seen as the basis of behavior than the "really existing" particles themselves. This is a very nice move on Hofstadter's part: he shows that the "macro", i.e., "I" level explanation of behavior is more to the point than a reductionist one. The metaphors and analogies he gives early in the book pay off very well here. It is worth reading his notion of "Thinkodynamics" in chapter 2 again.

In Chapter 8, Hofstadter finally defines the

*Strange Loop.*## Friday, June 10, 2011

### I Am a Strange Loop -- Commentary on Chapters 5-6

In chapter 5 Hofstadter relates his experience on what we could refer to as a "holiday with video feedback." He points a camera at a screen and makes images of images of images etc... and investigates the patterns that can emerge from this exercise. Some patterns "emerge" out of the systems, and persist, in a way that seems mysterious.

The infinitely rich video feedback he describes really represents a chaotic dynamical system. The dynamics is thought of as the light bouncing back and forth on patterns formed by previous bounces. In dynamical systems, the state of the system at time t+1 is dependent on its state at time t. Similarly, when you point a camera at a screen to get a video loop, whether a structure is present in the iteration of the loop at time t+1 is dependent on its placement at time t. Structures in the intersection of all iterations of the loop survive all iterations and become part of the final image. Other shapes may arise later on as pieces of the original image are bounced around. Hofstadter writes:

"It[a structure] wil not go away because it is forever refreshing itself, feeding on itself, giving rebirth to itself. Otherwise put, the emergent output pattern is a self-stabilizing structure whose origins, despite the simplicity of the feedback loop itself, are nearly impenetrable because the loop is cycled through so many times" (Hofstadter pp.98-99 Nookbook)

Sometimes dynamical systems, even though definable by simple relationships between time t and time t+1, can give rise to infinitely rich, and varied, structures; this is the basis of chaos theory. One phenomena that arises is infinite sensitivity to where you are at time t. That is, two points can be extremely close at some time t, but wander very far away from each other; one might end up going to the origin of an axis system, and another might blast out to infinity as time moves on. Now imagine what this means when you don't have infinite precision in measurement -- it means you cant tell which kind of point you've got!

The structures that emerge can be given names like "corridor" or "galaxy". His point is going to be that these structures, similar to what he says in chapters 2 and 3, should be regarded as fundamental explanatory units, rather than the individual photons that make up the image. Presumably, within consciousness one of the persistent structures is the self or "I".

In chapter 6 Hofstadter takes on the notion of the self. He says:

"Indeed to some people -- perhaps to most, perhaps even to us all -- the ineffable sense of being an "I" or a "first person", the intuitive sense of "being there" or simply "existing", the powerful sense of "having raw senations" (what some philosophers refer to as "qualia"), seem to be the realest things in their lives, and an insistent inner voice bridles furiously at any proposal that all this might be an illusion, or merely the outcome of some kind of physical processes taking place among "third person"(i.e., inanimate) objects. My goal here is to combat this strident inner voice." (Hofstadter pp. 100-101, Nookbook)

He brings up several ideas in these sentences and I'm going to try to separate them:

1. The sense of being an "I".

2. The Sense of having raw sensations(qualia)

3. That the self might be an illusion.

4. That the self is an outcome of processes taking place among "third person" objects.

1. From the wikipedia entry on Fichte:

"Fichte's account proceeds from th

e general principle that the I must set itself up as an individual in order to set itself up at all, and that in order to set itself up as an individual it must recognize itself as it were to a calling or summons (Aufforderung) by other free individual(s) — called, moreover, to limit its own freedom out of respect for the freedom of the other. The same condition applied and applies, of course, to the other(s) in its development. Hence, mutual recognition of rational individuals turns out to be a condition necessary for the individual 'I' in general. This argument for intersubjectivity is central to the conception of selfhood developed in the

In

Hofstadter, instead of offering ideas like the above, thinks in terms of a metaphorical camera turned toward the self. He imagines some organisms that can only point the camera outwards and so have no concept of the self, others, like humans, can twist and turn the camera toward the self and make all kinds of rich patterns in analogy with video feedback.

2. Hofstadter does not really address this as such, but thinks in terms of the necessity of symbols for having perceptions; that is, perceptions are always perceptions of something., not raw sensations. But he doesn't address the existence of "qualia" in this chapter.

3. That the self might be an illusion can be understood in a lot of different ways, a Buddhist way, for example. It can also be understood that since the self is not a "qualia" it doesn't exist, but is rather an inference from experience.

4. The self is the result of physical processess occuring among third person objects. He puts his finger here right on the problem. How does one move from the third person to the first person? I haven't been convinced so far that he addresses this.

The infinitely rich video feedback he describes really represents a chaotic dynamical system. The dynamics is thought of as the light bouncing back and forth on patterns formed by previous bounces. In dynamical systems, the state of the system at time t+1 is dependent on its state at time t. Similarly, when you point a camera at a screen to get a video loop, whether a structure is present in the iteration of the loop at time t+1 is dependent on its placement at time t. Structures in the intersection of all iterations of the loop survive all iterations and become part of the final image. Other shapes may arise later on as pieces of the original image are bounced around. Hofstadter writes:

"It[a structure] wil not go away because it is forever refreshing itself, feeding on itself, giving rebirth to itself. Otherwise put, the emergent output pattern is a self-stabilizing structure whose origins, despite the simplicity of the feedback loop itself, are nearly impenetrable because the loop is cycled through so many times" (Hofstadter pp.98-99 Nookbook)

Sometimes dynamical systems, even though definable by simple relationships between time t and time t+1, can give rise to infinitely rich, and varied, structures; this is the basis of chaos theory. One phenomena that arises is infinite sensitivity to where you are at time t. That is, two points can be extremely close at some time t, but wander very far away from each other; one might end up going to the origin of an axis system, and another might blast out to infinity as time moves on. Now imagine what this means when you don't have infinite precision in measurement -- it means you cant tell which kind of point you've got!

The structures that emerge can be given names like "corridor" or "galaxy". His point is going to be that these structures, similar to what he says in chapters 2 and 3, should be regarded as fundamental explanatory units, rather than the individual photons that make up the image. Presumably, within consciousness one of the persistent structures is the self or "I".

In chapter 6 Hofstadter takes on the notion of the self. He says:

"Indeed to some people -- perhaps to most, perhaps even to us all -- the ineffable sense of being an "I" or a "first person", the intuitive sense of "being there" or simply "existing", the powerful sense of "having raw senations" (what some philosophers refer to as "qualia"), seem to be the realest things in their lives, and an insistent inner voice bridles furiously at any proposal that all this might be an illusion, or merely the outcome of some kind of physical processes taking place among "third person"(i.e., inanimate) objects. My goal here is to combat this strident inner voice." (Hofstadter pp. 100-101, Nookbook)

He brings up several ideas in these sentences and I'm going to try to separate them:

1. The sense of being an "I".

2. The Sense of having raw sensations(qualia)

3. That the self might be an illusion.

4. That the self is an outcome of processes taking place among "third person" objects.

1. From the wikipedia entry on Fichte:

"Fichte's account proceeds from th

e general principle that the I must set itself up as an individual in order to set itself up at all, and that in order to set itself up as an individual it must recognize itself as it were to a calling or summons (Aufforderung) by other free individual(s) — called, moreover, to limit its own freedom out of respect for the freedom of the other. The same condition applied and applies, of course, to the other(s) in its development. Hence, mutual recognition of rational individuals turns out to be a condition necessary for the individual 'I' in general. This argument for intersubjectivity is central to the conception of selfhood developed in the

*Doctrine of Science*(aka 'Wissenschaftslehre'). In Fichte's view consciousness of the self depends upon resistance or a check by something that is understood as not part of the self yet is not immediately ascribable to a particular sensory perception."In

*Discipline and Punish,*Michel Foucault says that the self is the internalization of the modes of social control.Hofstadter, instead of offering ideas like the above, thinks in terms of a metaphorical camera turned toward the self. He imagines some organisms that can only point the camera outwards and so have no concept of the self, others, like humans, can twist and turn the camera toward the self and make all kinds of rich patterns in analogy with video feedback.

2. Hofstadter does not really address this as such, but thinks in terms of the necessity of symbols for having perceptions; that is, perceptions are always perceptions of something., not raw sensations. But he doesn't address the existence of "qualia" in this chapter.

3. That the self might be an illusion can be understood in a lot of different ways, a Buddhist way, for example. It can also be understood that since the self is not a "qualia" it doesn't exist, but is rather an inference from experience.

4. The self is the result of physical processess occuring among third person objects. He puts his finger here right on the problem. How does one move from the third person to the first person? I haven't been convinced so far that he addresses this.

## Wednesday, June 8, 2011

### I Am A Strange Loop -- Commentary on Chapter 4

"But science, spurred by its powerful illusion, speeds irresistably toward its limits where its optimism, concealed in the essence of logic, suffers shipwreck. For the periphery of the circle of science has an infinite number of points; and while there is no telling how this circle can ever be portrayed completely, noble and gifted men nevertheless reach, e'er half their time and inevitably, such boundary points on the periphery from which one gazes into what defies illumination. When they see to their horror how logic coils up at these boundaries and finally bites its own tail -- suddenly the new form of insight breaks through,

-- Friedrich Nietzsche

I think about that experience often; I use it as a metaphor for intellectual experiences I sometimes have -- especially in mathematics. When I come to understand something surprising in math that violates my intuition, or seems mysterious to me, and of course the experience can be really cool. But sometimes, when I've had to solve some problem or other, and there is no one I can turn to for the answer, the experience takes on a sinister, confidence undermining aspect as well.

I wonder how Bertrand Russell felt when he realized his now famous paradox: "The set of all sets that do not contain themselves." Call this set S. Now, is S a member of itself? Well, if S is a member of itself, then it must satisfy the definition of S, which is not being a member of itself. Thus, if S is a member of itself, then it is not a member of itself. What? Has logic coiled back on you yet? No? OK, suppose S is not a member of itself, then it satisfies its own definition, thus it IS a member of itself. So, if it isn't a member of itself, then it is a member of itself. Down the runny toilet(see chapter 4) goes grounding all arithmetic in set theory!

Hofstadter says that Russell was too timid when confronted with this problem, actually Hofstadter is right here, by eliminating all self-reference:

"This trauma instilled in him a terror of theories that permitted loops of self-containment or of self-reference, since he attributed the intellectual devastation he had experienced to loopiness and to loopiness alone."(Hofstadter, pg.88 Nookbook)

Remember that Russell and Whitehead worked for years on this project. I think Hofstadter's a little hard on him when he says: "...I was disappointed for a lifetime with the oncebitten twice-shy mind of Bertrand Russell."(Hofstadter pg. 89 Nookbook).

It is this type of paradox that Goedel later exploits in his famous incompleteness theorem. I will be going into this stuff when I get to that chapter. I'm taking this opportunity to read the original paper closely. But I'll say here that the goal reducing all of mathematics to formal systems -- perhaps a hubristic goal -- was shown impossible. Perhaps then there is indeed something tragic here. NO, we cannot take all knowledge and completely represent it in a single formal system because we can't even do it with arithmetic!

*tragic insight*which, merely to be endured, needs art as a protection and remedy."-- Friedrich Nietzsche

*The Birth of Tragedy**When I was 12 I got a pair of flippers and goggles. One day I got into the water of this dock where I could stand. I decided to do the backstroke out about 50 feet. I then made the mistake of looking down. Looking through the clear water with my goggles I saw a beam of light enter the water, and go deeper ... and deeper .. and deeper, and fade ever so slowly until it was finally completely absorbed by the water; I could tell this was nowhere near the bottom. Immediately my stomach practically dropped right out of me. I panicked, flailing my legs as fast as I could to get back to the shore. I will never forget that experience as long as I live.*I think about that experience often; I use it as a metaphor for intellectual experiences I sometimes have -- especially in mathematics. When I come to understand something surprising in math that violates my intuition, or seems mysterious to me, and of course the experience can be really cool. But sometimes, when I've had to solve some problem or other, and there is no one I can turn to for the answer, the experience takes on a sinister, confidence undermining aspect as well.

I wonder how Bertrand Russell felt when he realized his now famous paradox: "The set of all sets that do not contain themselves." Call this set S. Now, is S a member of itself? Well, if S is a member of itself, then it must satisfy the definition of S, which is not being a member of itself. Thus, if S is a member of itself, then it is not a member of itself. What? Has logic coiled back on you yet? No? OK, suppose S is not a member of itself, then it satisfies its own definition, thus it IS a member of itself. So, if it isn't a member of itself, then it is a member of itself. Down the runny toilet(see chapter 4) goes grounding all arithmetic in set theory!

Hofstadter says that Russell was too timid when confronted with this problem, actually Hofstadter is right here, by eliminating all self-reference:

"This trauma instilled in him a terror of theories that permitted loops of self-containment or of self-reference, since he attributed the intellectual devastation he had experienced to loopiness and to loopiness alone."(Hofstadter, pg.88 Nookbook)

Remember that Russell and Whitehead worked for years on this project. I think Hofstadter's a little hard on him when he says: "...I was disappointed for a lifetime with the oncebitten twice-shy mind of Bertrand Russell."(Hofstadter pg. 89 Nookbook).

It is this type of paradox that Goedel later exploits in his famous incompleteness theorem. I will be going into this stuff when I get to that chapter. I'm taking this opportunity to read the original paper closely. But I'll say here that the goal reducing all of mathematics to formal systems -- perhaps a hubristic goal -- was shown impossible. Perhaps then there is indeed something tragic here. NO, we cannot take all knowledge and completely represent it in a single formal system because we can't even do it with arithmetic!

## Sunday, June 5, 2011

### I Am A Strange Loop -- Chapters 2 and 3

In Chapter 2, Hofstadter begins his assault on pure reductionism, and also on John Searle(I also recommend the wikipedia page on John Searle). Hofstadter emphasizes that reductionism as he defines it, reducing all explanation to the microphysical, is not the best way to go about understanding consciousness. He believes that consciousness is the result of many, many, neurons firing in just the right way. He gives a lot of examples of how it is better in some contexts to explain things in terms of collections rather than in terms of the individual particles that make up a system.

While discussing the explanatory superiority of collectives over individuals, he states further that abstractions, like "dog"(i.e.,concepts within consciousness), are the proper objects of study for really understanding the brain, which he describes as a thinking organ. But I think sometimes he gets collections and conscious ideas tangled up a bit. Yes, I can see how patterns of neural activity may be associated with the thought "dog", and that it is better to think in terms of "dog" sometimes than the associated neural pattern, but sometimes I get the feeling that he slips between collections and conscious abstractions without keeping track of the distinction properly.

There is a lack of appreciation of the explanatory gap between physical things, like collections of neurons, and conscious entities, like the idea "dog", that Levine talks about in the article I mention in a previous post. By the way, telling me the whole is more than the sum of its parts doesn't cut it; I need more than that. There is, simply put, a qualitative difference between mental abstractions and physical collectives that Hofstadter doesn't satisfy me about here. As Levine says:

"When I conceive of the mental, it seems utterly unlike the physical. Antimaterialists insist that from this intuitive difference we can infer a genuine metaphysical difference. Materialists retort that the nature of reality, including the ultimate natures of its constituents, is a matter for discovery; an objective fact that cannot be discerned a priori."

This passage is much more probative regarding the difference between the mental and physical than anything Hofstadter says here(vividly though he puts it). Levine concludes:

"The explanatory gap argument doesn't demonstrate a gap in nature, but a gap in our understanding of nature. Of course a plausible explanation for there being a gap in our understanding of nature is that there is a genuine gap in nature. But so long as we have countervailing reasons for doubting the latter, we have to look elsewhere for an explanation of the former. "

Thus, while there is an explanatory gap, a point with which I completely agree, one can't actually conclude a non-physical cause from it.

Then Hofstadter takes on John Searle -- I gather from the tone this is not their first run-in. Searle emphasizes that consciousness is not just the result of patterns, but depends also on the causal properties of the materials out of which the consciousness is constructed: a replica of a human being made out of yarn or beer cans would not have consciousness because yarn and beer cans do not have the right causal properties. Hofstadter just goes off!

"...because John Searle has a gift for catchy imagery, his specious ideas have, over the years, had a great deal of impact on many professional colleagues, graduate students, and lay people."

Interesting that Hofstadter is also known for his vivid writing style. He also uses vivid imagery in chapters 2 and 3 to describe his primary idea. He asserts that John Searle defeats a straw argument that Hofstadter et al assert that a single neuron has consciousness. Hofstadter goes to great lengths to explain that consciousness does not reside in a single neuron but in the collective. I frankly do not find that this move accomplishes anything. I am no more convinced that consciouness somehow pops out of the collective than out of the individual. The gap is just too big.

P.S. -- For fun, check out the wikipedia page on supervenience.

While discussing the explanatory superiority of collectives over individuals, he states further that abstractions, like "dog"(i.e.,concepts within consciousness), are the proper objects of study for really understanding the brain, which he describes as a thinking organ. But I think sometimes he gets collections and conscious ideas tangled up a bit. Yes, I can see how patterns of neural activity may be associated with the thought "dog", and that it is better to think in terms of "dog" sometimes than the associated neural pattern, but sometimes I get the feeling that he slips between collections and conscious abstractions without keeping track of the distinction properly.

There is a lack of appreciation of the explanatory gap between physical things, like collections of neurons, and conscious entities, like the idea "dog", that Levine talks about in the article I mention in a previous post. By the way, telling me the whole is more than the sum of its parts doesn't cut it; I need more than that. There is, simply put, a qualitative difference between mental abstractions and physical collectives that Hofstadter doesn't satisfy me about here. As Levine says:

"When I conceive of the mental, it seems utterly unlike the physical. Antimaterialists insist that from this intuitive difference we can infer a genuine metaphysical difference. Materialists retort that the nature of reality, including the ultimate natures of its constituents, is a matter for discovery; an objective fact that cannot be discerned a priori."

This passage is much more probative regarding the difference between the mental and physical than anything Hofstadter says here(vividly though he puts it). Levine concludes:

"The explanatory gap argument doesn't demonstrate a gap in nature, but a gap in our understanding of nature. Of course a plausible explanation for there being a gap in our understanding of nature is that there is a genuine gap in nature. But so long as we have countervailing reasons for doubting the latter, we have to look elsewhere for an explanation of the former. "

Thus, while there is an explanatory gap, a point with which I completely agree, one can't actually conclude a non-physical cause from it.

Then Hofstadter takes on John Searle -- I gather from the tone this is not their first run-in. Searle emphasizes that consciousness is not just the result of patterns, but depends also on the causal properties of the materials out of which the consciousness is constructed: a replica of a human being made out of yarn or beer cans would not have consciousness because yarn and beer cans do not have the right causal properties. Hofstadter just goes off!

"...because John Searle has a gift for catchy imagery, his specious ideas have, over the years, had a great deal of impact on many professional colleagues, graduate students, and lay people."

Interesting that Hofstadter is also known for his vivid writing style. He also uses vivid imagery in chapters 2 and 3 to describe his primary idea. He asserts that John Searle defeats a straw argument that Hofstadter et al assert that a single neuron has consciousness. Hofstadter goes to great lengths to explain that consciousness does not reside in a single neuron but in the collective. I frankly do not find that this move accomplishes anything. I am no more convinced that consciouness somehow pops out of the collective than out of the individual. The gap is just too big.

P.S. -- For fun, check out the wikipedia page on supervenience.

## Saturday, June 4, 2011

### I Am a Strange Loop -- Commentary on Chapter 1 part 2

I've been thinking about experience and qualia and I can't see a way around the conclusion that our current science does not bridge the gap Levine talks about in the essay I gave a reference to a couple of posts ago. One doesn't need to follow the entire gobbledy-gook of his argument to see that there is something about subjective experience that, from an explanatory point of view, is missing from physical explanations. Even though I believe that mental phenomena are ultimately dependent on the physical(I am prone to optical migranes, have been on a variety of medications that have affected my mental state, am affected reliably by caffeine, etc.. and have concluded that if the physical substrates of my body are changed, so is the mental, with the inference also that when I die my mental experience will cease) it seems clear to me that there is an explanatory gap when it comes to reducing my experience of the color red to some combination of the Strong, Electro-Weak, and Graviational forces, none of which have anything like bridge concepts between description of physical reality and my qualia.

This does not mean that there must be some separable non-physical reality, perhaps the mental is some sort of dimension to physical we can't uncover; physical things in certain combination also combine parts of the mental dimension in ways to give a mental experience. When the physical organization of the nervous system disintegrates, the mental life disintegrates with it.

Thus I think that there is something to arguments that from the fact that I can conceive of philosophical zombies I can conclude there is an explanatory gap between the physical and mental, but I do not think that the mental hovers independently of the physical altogether. You really should read the Chalmers site on philosophical zombies, the others are a little more optional.

Back to Hofstadter... He talks about the amount of "souledness" an organism has and asks:

"Why should there not likewise be an average degree of souledness for adults[100], plus a wide range around that average, maybe (as for IQ), going as high as 150 or 200 hunekers[Hofstadter's fictional scale of consciousness, or "souledness"] in rare cases, and down to 50 or lower in others?"(Hofstadter Nookbook pp. 45-46).

He realizes this can cause some people to become a little squeemish morally, but continues anyway. He states that just-fertilized human eggs have no hunekers worth of soul, and then says the same regarding 5 month fetuses. I wonder if Harris(see his

The moral consequences here are obvious, especially for those suffering from mental disability.

This does not mean that there must be some separable non-physical reality, perhaps the mental is some sort of dimension to physical we can't uncover; physical things in certain combination also combine parts of the mental dimension in ways to give a mental experience. When the physical organization of the nervous system disintegrates, the mental life disintegrates with it.

Thus I think that there is something to arguments that from the fact that I can conceive of philosophical zombies I can conclude there is an explanatory gap between the physical and mental, but I do not think that the mental hovers independently of the physical altogether. You really should read the Chalmers site on philosophical zombies, the others are a little more optional.

Back to Hofstadter... He talks about the amount of "souledness" an organism has and asks:

"Why should there not likewise be an average degree of souledness for adults[100], plus a wide range around that average, maybe (as for IQ), going as high as 150 or 200 hunekers[Hofstadter's fictional scale of consciousness, or "souledness"] in rare cases, and down to 50 or lower in others?"(Hofstadter Nookbook pp. 45-46).

He realizes this can cause some people to become a little squeemish morally, but continues anyway. He states that just-fertilized human eggs have no hunekers worth of soul, and then says the same regarding 5 month fetuses. I wonder if Harris(see his

*The Moral Landscape*) has considered hunekers in his scientific view of morality? Clearly Hofstadter connects hunekers with our moral obligation to entities since he mentions he has no problem eating tomatoes since they have no hunekers(the store-bough brand I trust; home-grown tomatoes clearly have a soul!), or killing mosquitoes, more problems killing ants, etc..The moral consequences here are obvious, especially for those suffering from mental disability.

## Wednesday, June 1, 2011

### I Am a Strange Loop commentary Chapter 1 -- part 1

"A Mote it is to trouble the mind's eye" -- Hamlet

Here's another one:

"Thoughts beyond the reaches of our souls" -- Hamlet

One of my favorite parts of "Men In Black" is when the alien is talking about human beings and he calls us "barely conscious pond scum". The alien clearly believes he has no reason to respect life forms that possess comparatively impoverished levels of consciousness. Ironically, this alien is actually a bug, a form of life humans think of as barely conscious -- unless you're in a Kafka story. Later, Tommy Lee Jones says that human thought is considered a form of disease by the rest of the galaxy.

In Chapter 1 of

On the other hand, Hofstadter recounts how he fainted at the prospect of taking a guinea pig up to be killed for physiological research during his student days. If he faints for gunea pigs, I'm not too worried he'll be insensitive to humans lower on the consciousness scale. This is not necessarily a valid inference: Hemingway believed that folks who were very concerned about animals were often not very sensitive to people. For the record, I think Hofstadter is probably a nice guy who goes out of his way not to hurt any sentient being, human or not. Recall that, I think it was Bentham who said this, that our obligation toward a living thing is in proportion to its ability to SUFFER. You know, they say men have a lower threshold of pain than women, so my wife should be especially nice to me.

Such ideas are, as you know, no where near being new. From the Gold, Silver, and Bronze souls of Plato's

Well, what is consciousness anyway, and how do we know how much of it we have? Is it how many chunks we can hold in conscious memory at one time, it's supposed to be somewhere between 5 and 7?

Or is it a matter of the quality of things that can pass into consciousness, for example, a mental picture somehow of the rich layers of a passage of Chopin? Or perhaps it's an area computed between both quantity and quality -- OK, now I'm being silly. One way of putting this is: how many varieties of qualia can a given sentient being have at a time and from how big a list of possible qualia can this being choose? We'll get into this a little in the next post.

Here's another one:

"Thoughts beyond the reaches of our souls" -- Hamlet

One of my favorite parts of "Men In Black" is when the alien is talking about human beings and he calls us "barely conscious pond scum". The alien clearly believes he has no reason to respect life forms that possess comparatively impoverished levels of consciousness. Ironically, this alien is actually a bug, a form of life humans think of as barely conscious -- unless you're in a Kafka story. Later, Tommy Lee Jones says that human thought is considered a form of disease by the rest of the galaxy.

In Chapter 1 of

*I am a Strange Loop*, the most controversial, and to some troubling, thing Hofstadter does is arrange sentient beings on a scale of consciousness, what's more, he ranks normal adults higher than retarded people, children, people with Alzheimer's disease etc... This has caused one blogger to wonder whether there is something, well, politically dangerous, about doing so(http://newskeptic.blogspot.com/2007/11/i-am-strange-loop-consciousness.html).On the other hand, Hofstadter recounts how he fainted at the prospect of taking a guinea pig up to be killed for physiological research during his student days. If he faints for gunea pigs, I'm not too worried he'll be insensitive to humans lower on the consciousness scale. This is not necessarily a valid inference: Hemingway believed that folks who were very concerned about animals were often not very sensitive to people. For the record, I think Hofstadter is probably a nice guy who goes out of his way not to hurt any sentient being, human or not. Recall that, I think it was Bentham who said this, that our obligation toward a living thing is in proportion to its ability to SUFFER. You know, they say men have a lower threshold of pain than women, so my wife should be especially nice to me.

Such ideas are, as you know, no where near being new. From the Gold, Silver, and Bronze souls of Plato's

*Republic,*to The Great Chain of Being of the Elizabethan World Picture, to Hegel's theory, to social constructionist epistemologies, people are always qualifying or even ordering consciousnesses. Hofstadter invents a scale named for a commentator, Huneker, on Chopin who mentioned that not everyone has a big enough soul to understand and play Chopin.Well, what is consciousness anyway, and how do we know how much of it we have? Is it how many chunks we can hold in conscious memory at one time, it's supposed to be somewhere between 5 and 7?

Or is it a matter of the quality of things that can pass into consciousness, for example, a mental picture somehow of the rich layers of a passage of Chopin? Or perhaps it's an area computed between both quantity and quality -- OK, now I'm being silly. One way of putting this is: how many varieties of qualia can a given sentient being have at a time and from how big a list of possible qualia can this being choose? We'll get into this a little in the next post.

### I Am a Strange Loop -- Preliminaries part 2

I followed-up and found some really cool websites.

For Philosophical Zombies see

http://www.consc.net/zombies.html

For more on Qualia see:

http://cognet.mit.edu/posters/TUCSON3/Levine.html

and for Daniel Dennett see

http://ase.tufts.edu/cogstud/papers/quinqual.htm

For Philosophical Zombies see

http://www.consc.net/zombies.html

For more on Qualia see:

http://cognet.mit.edu/posters/TUCSON3/Levine.html

and for Daniel Dennett see

http://ase.tufts.edu/cogstud/papers/quinqual.htm

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