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Waiting for Godot -- Reading Godot, the conglomerative effect

      The heart of Lois Gordon's book is her interpretation of the various features of the play in terms of Freudian conglomeration.  She says: "Freud uses the term conglomeration in the process of collecting the fragmentary components of the dream.  This is a concept akin to the mental operation of "secondary revision", which gives final shape and form to the compressed dream image.  In following Freud's procedure, I shall speak of the conglomerative effect or conglomerative refrain in order to indicate what traditionally would be called the dominant theme of the play, always in the form of thesis and antithesis."(pg. 75) Thus, we see the various verbal formulations, including the ambiguous or self-contradictory speeches, dialogues, stage directions(when compared with the dialogue), as if they were disorganized components of a dream.  The assemblage of a theme is therefore analogous to the assembling of a narrative and meaning among the pieces of a drea

Waiting for Godot -- 'Reading Godot' by Lois Gordon, entry 2

"The power of Waiting for Godot derives from its exposure of the emotional life in counterpoint to the existential condition -- Beckett's revelation of the unconscious feelings that accompany the quest for salvation in a world bereft of meaning. Godot's much-repeated 'Let's go.(they do not move)' epitomizes not just the tension of individual action in a meaningless world; it also demonstrates the limits of the will set against the contraining and deterministic forces of a controlling psyche."(pg. 70-71) Existentialism without the freedom.   She asserts the influence of Freudianism on Beckett all the way down to the set design, props, the direction the characters walk and how they move about on the stage: "It is as though Beckett were weaving into his existential landscape a map of the unconscious world, again, either as it functions simultaneously with rational thinking or is transformed into a dreamscape...Each object, word, and movement ultimatel

Waiting for Godot -- "Reading Godot" by Lois Gordon part 1 of many

"Werner Heisenberg and quantum physics, no less than Freud and Sartre, have demonstrated that the limits of our universe are determined by the limits of our measuring instruments, whether they are atomic clocks, blood pressure cups, or nouns and verbs (Gordon pg. 18) I usually dislike it when humanities' types bring in physics to illustrate some point about relativism etc..., but I make an exception in this case.  It is true that there is a fundamental limit to our ability to locate and measure the momentum of particles precisely, and that ultimately science is about what we can measure, and there are apparently limitations on that, and thus on our ability to know.  We spin our theories based on what we can possibly measure. Perhaps we assume that the limitations of what we can measure are the limitations of the universe itself, an obvious intellectual error, but just as there are undoubtedly aspects of the physical universe beyond our ability to measure, there are perhaps t

Waiting for Godot and Myth of Sisyphus part 2

     I will continue an examination of Waiting for Godot by introducing a selection from the brilliant book, Reading Godot, by Lois Gordon.  When the actual book arrives from Amazon I will read over everything she has to say.  Gordon begins this section with "In a world devoid of belief systems, the mind and heart cry out for validation, for the assurance that life has meaning and actions have purpose.  One may accept, as an existential truth, the assumption that despite the individual's endeavors to comprehend or change the world, "there is no new thing under the sun"(Ecclesiastes), and, as Beckett puts it, "the tears of the world are of a constant quantity..."  But one also occupies a world of temporal measurement.  Time passes and one ages, and, facing these inescapable facts, one journeys with tenacious will through the arbitrary divisions of time and space holding onto goals and belief systems as if they were absolute(Bloom pg. 124) We inhabit both

Waiting For Godot and the Myth of Sisyphus part 1

     I bought the Harold Bloom 'Modern Critical Interpretations' edition for 'Waiting for Godot' and read several of the essays.  All of the essays mention Albert Camus's short essay, 'The Myth of Sisyphus', so I figured I would start by looking over this essay and quoting what some of the interpreter's say; this is an essay I first read in high school and it was a poignant experience to read it again as I remember the effect the essay had on me all those years ago.      One of the first things I noticed when I got my old copy of the essay was an underlining I had made in the essay;  I would have done this some time around 1982: "But when he had seen again the face of this world, enjoyed the water and sun, warm stones, and the sea, he no longer wanted to go back to the infernal darkness."(pg. 88-89) When I recall the bouts with depression, the sense of futility, along with the melodramas of my adolescence, this line, with its emphasis on si

Beckett!

My next series of articles will be explorations of the dramatic works of Samuel Beckett.  I have read some of his plays and he is becoming one of my favorite author.  So get ready for the ABSURD.  I will start next time with Waiting for Godot.

Derek Parfit -- The Final Entry

After a long, tedious, explanation of what is wrong with various versions of "Subjective Theories", without really dealing with what I would call the 'real problem', nihilism, he talks about Kant. I have to say that I find actually reading the text unbearable.  So, I'm going back to the summary to see if he gives me an argument I can sink my teeth into. So, after skimming through much more tedium I  violated all canons of what is good and decent by going right to the "conclusions" section! Sigh, and here is what I was treated to: "everyone ought always to do whatever would make things go best."(pg. 74)   Let's read that again: "everyone ought always to do whatever would make things go best."(pg. 74)  I need a 600 page book for this? Here's another gem: "When there is only one set of principles that everyone could rationally will to be universal laws, these are the only principles, we can argue, that no one could re

Derek Parfit -- Finally saying something

I suppose it's not surprising that it's easier to poke holes in the opposition's theories than it is to put forward one of your own; that's one reason why the next chapter, "Subjective Theories", is better.  He gives the example of running away from a snake in the mistaken belief that running will save your life(you should stand still).  He says: "Subjectivists might claim that (A) reasons are provided only by desires that depend on true beliefs, You have no reason to run away, (A) implies, because your desire depens on the false belief that this act would save your life".(pg. 107) He then defines the Telic Desire Theory: "We have most reason to do whatever would best fulfil or achieve our present telic desires or aims."(pg. 108) He then says that the problem here is that sometimes our telic desires are based on false beliefs. Thus we have "the Error-Free Desire Theory: We have most reason to do whatever would best fulfil or

Derek Parfit --- Objective Theories and unfree desires

Parfit provides an unilluminating distinction between subjective and objective theories: On many subjective theories, the strength of the these reasons depends on the strength of these desires or on our preferences.  On objective theories, the strength of these reasons depends instead on how good, or worth achieving, the fulfilment of these desires would be. Many of us often have stronger desires for what would be less worth achieving."(pg. 95) Is this guy my Sunday School teacher? He then reiterates the tired formula that we do not choose what we desire etc....; this was known in the 19th century as the "law of motivation", at least is was for Schopenhauer.   He say "we can choose which desires to adopt as aims, and try to fulfil."  What makes him think this?? The book reads like he's writing right off the top of his head, which is not a good thing in this case. "Our desires are rational, I have claimed, when we want events whose features giv

Derek Parfit -- Reasons weighed against Reasons

Parfit goes on at some length about how facts give us reasons.  He gives examples like being allergic to walnuts gives us a reason not to eat them.  He goes on to say that reasons counteract other reasons.  None of this is particularly interesting.  He praises rationality as a way of determining how we ought to act.  He goes on at some tedious length about 'relevant' reasons providing 'sufficient' reasons for acting in certain ways, all the while begging the questions that plague the history of ethics -- at least, so far.  My advice is that if he's going to re-establish normative ethics on some sort of foundation that people can really sink their teeth into, he needs to get to it a tad sooner otherwise people will think he doesn't really have anything. Here goes: "When we call something good, in what we call the reason-implying sense, we mean roughly that there are certain kinds of fact about this thing's nature, or properties, that would in certai

Derek Parfit -- a few comments

The fundamental problem Parfit is going to have is the idea that there is some "objective" content to facts that yield proper beliefs and, from that, actions.  Facts, assuming they can be formulated in an objective way, which is not exactly a postmodern position, must then additionally yield objective implications.  Even those who believe in objective facts may have problems with the notion that there are objective moral implications.  Look, either there are or there aren't objective implications.  If there are, then they are valid for all people, irrespective of a person's cultural background etc..., if there aren't, then the interpretations of facts are culturally relative.  And this is assuming that the facts themselves have objective content. Of course, these days people these days say that facts are "value-laden" -- this is a way of describing the cultural relativity in the depiction of the "objective" world itself.  The advantage of thi

Derek Parfit Chapter 1

Parfit starts with the following: "We are the animals that can both understand and respond to reasons. Facts give us reasons when they count in favour of our having some belief or desire, or acting in some way."(pg.50) I love this opening!  It summarizes a take on that which is distinctly human, that corresponds to our ability to do science as well as to consider ethics.  We move from facts to beliefs and from beliefs to rationality: "Though it is facts that give us reasons, what we can rationally want or do depends instead on our beliefs." Facts are already not something separate from beliefs and morality; they provide us with "reasons". (pg. 50) How can a fact provide us with reasons? Only if facts are already constructed by some reasoning faculty, perhaps.  Facts don't just hang out there neutrally.  If they did, nothing could be done with them; they cannot be said to 'count' for or against anything. The notion that there is this gulf

Derek Parfit's raison d'becoming a philosopher

I love the following from the beginning of On What Matters: "My debts to Sidgwick are easy to describe.  Of my reasons for becoming a graduate student in philosophy, one was the fact that, in wondering how to spend my life, I found it hard to decide what really matters.  I knew that philosophers tried to answer this question, and to become wise.  It was disappointing to find that most of the philosophers who taught me, or whom I was told to read, believed that the question 'What matters?' couldn't have a true answer, or didn't even make sense.  But I bought a second-hand copy of Sidgwick's book, and I found that he at least believed that some things matter"(pg. 41) This makes me like Parfit a lot, maybe he and Sidgwick can convince me that something matters, that would be cool.

Derek Parfit entry 1

  I don't know too much about Derek Parfit yet but he seems to want to combine Kantianism and consequentialism, at least according to the introduction.  He also seems to think that the "Formula of Universal Law" can be saved in a form like "Everyone ought to follow the principles whose universal acceptance everyone could rationally will."(pg. 22)    Without having read the guts of Parfit's arguments I can say at the outset that I'm going to have difficulty accepting this kind of moral foundation.  I'm not convinced that I have to behave in a way that any "rational being" would will.  I don't think that there is a rational being, or anything such as rationally willing something.  Parfit has chapters on Nietzsche, and perhaps he deals with this issue, but I don't see how Parfit is going to be successful.  If I don't believe in rationality, and I don't believe in rationally willing anything, I don't see how I'm going

Derek Parfit

OK, so the Blackburn, the James, and Snow Crash turned out not to be interesting enough to hold my attention.  So, I've purchased On What Matters, in two volumes, by Derek Parfit.  I'm hoping this will keep my interest and I'll have some entries on it.

Pragmatism -- Entry 3, the practicality of God

James takes on traditional philosophical problems such as the meaning of matter. Matter as a substance underlying experience.  What does it mean to say that the world is constituted by such?  Philosophers later in the century would say that such metaphysical questions are meaningless.  James has a different approach.  He says first: "What do we MEAN by matter? What practical difference can it make NOW that the world should be run by matter or by spirit? I think we find that the problem takes with this a rather different character...It makes not a single jot of difference so far as the PAST of the world goes, whether we deem it to have been  the work of matter or whether we think a divine spirit was its author..."(pg. 41) It sounds like he might agree with those later thinkers when he says: "The pragmatist must conclude consequently say that the two theories, in spite of their different-sounding names, mean exact same thing, and that the dispute is purely verbal.&qu

Pragmatism -- Entry 2

     James begins the second lecture by describing "The Pragmatic Method": "Is the world one or many? -- fated or free? -- material or spiritual? -- here are notions either of which may or may not hold good of the world; and disputes over such notions are unending. ...What difference would it practically make to anyone if this notion rather than that notion were true?  If no practical difference whatever can be traced, then the alternatives men practically the same thing, and all dispute is idle."(pg. 21) This is typical of the early 20th century obsession with eliminating metaphysics from discourse.  This will turn into a full-scale neurosis by the middle of the century.  This is James' response.  It has the benefit of not ending up in yet another round of disputes over underdetermination, falsification, verification, etc.. -- at least not so far.  "The whole function of philosophy ought to be to find out what definite difference it will make to you

Pragmatism -- Entry 1

     James opens the book by describing the state of philosophy as he sees it -- in 1906.  He contrasts the scientific, atheist, materialist type of thinker with the rationalist, religious type.  He identifies empiricist philosophy with thinkers like Herbert Spencer, who was widely read at the end of the 19th century but is not so now.  On the other hand we get idealists like the "Anglo-Hegelians" whose philosophies have nothing to the world in which we live.        About the empiricists he says: "Never were as many men of decidedly empiricist proclivity in existence as there are at the present day.  Our children, one may say, are almost born scientific.  But our esteem for facts has not neutralized in us all religiousness.  It is itself almost religious.  Our scientific temper is devout"(pg. 9) He goes on: "The romantic spontaneity and courage are gone, the vision is materialistic and depressing.  Ideals appear as inert by-products of physiology; wha

The Will to Believe by William James

     William James produces a very strident response to William Clifford, and one that sounds much more modern to my ears.  The argument may be over religion, but it applies to much more than that.  James' essay is a rejection of the straight enlightenment program; it sounds more Nietzschean, more relativistic, and, frankly, more correct to me.  Consider the following remark in his introduction to this collection of essays: "Something -- 'call it fate, chance, freedom, spontaneity, the devil, what you will" -- is still wrong and other and outside and unincluded, from your point of view, even though you be the greatest of philosophers.  Something is always mere fact and givenness; and there may be in the whole universe no one point of view extant from which this would not be found to be the case."(pg. 2) This could have been written in the last 50 years. Regarding The Will to Believe, he is offering a "defence of our right to adopt a believing atitude i

The Ethics of Belief; The Ethics of Religion by William Clifford

     So I've read "The Ethics of Belief" and "The Ethics of Religion" by the fine mathematician William Clifford; I've also read "The Will to Believe" by the great William James.  William Clifford is a typical science-worshipper who says we have an obligation to believe only those things that stand the test of evidence.  While he has some excellent turns of phrase in "The Ethics of Belief", it is not that interesting on the whole.  "The Ethics of Religion" is more interesting since he is taking on religion directly and letting it rip.  I have to say I found Clifford's approach to knowledge naive and a tad boring.  I found William James to be much more interesting; this is not to say I think James is necessarily right, but he is more interesting(I will deal with William James in the next post).      The most interesting ideas Clifford has in "The Ethics of Belief" is how our web of ideas can be infected by ideas p

Truth: A Guide, by Simon Blackburn, William Clifford vs William James

     In Chapter 1, Blackburn sets up the oppositon in views between William Clifford and William James; specifically starting with Clifford's "The Ethics of Belief" and William James' famous response, "The Will to Believe".  Blackburn comes down pretty hard on the side of William Clifford.  Blackburn, I think, will use this to establish his core positions against relativism:      "And, of course, Clifford is right.  Someone sitting on a completely unreasonable belief is sitting on a time bomb.  The apparently harmless, idiosyncratic belief of the Catholic Church tht one thing may have the substance of another, although it displays absolutely none of its empirical qualities, prepares people for the view that some people are agents of Satan in disguisem which in turn makes it reasonable to destroy them.  Clifford also emphasizes our social duty,  Our beliefs help to create the world in which our descendants will live.  Making ourselves gullible or credu

Truth: A Guide, by Simon Blackburn, entry 1

"If truth is thought of as a goal that can never be attained, those who rather conspicuously do not care much about it will seem that much less villainous than they are." (pg. xvi) In his Introduction, sets up the opposition between absolutists on the one side, and relativists on the other.  He masterfully lays out what he means by each.  Along the way there are some real gems, for example: "[William] James describes the absolutist as having a religious temperament, whether the object of his religion is some traditional text or diety, or a new one, such as the Market, or Democracy, or Science.  This may also seem surprising, since religious lives can be full of doubt and worry and dark nights of the soul, and... in the modern world it is the relativists as much as the absolutists who belong to the cults."(xvii) Lest you think that absolutism only reigns in the domain of religion, recall we may make many other commitments with the same fervor; in fact, we mi

Simon Blackburn and Neal Stephenson

       Let me say this: both Simon Blackburn and Neal Stephenson are awesome!  From looking at Truth: A Guide, by Blackburn, the title being a reference to Maimonides(again, see the wonderful wikipedia page), and Snow Crash, by Stephenson, these are the two best books I've looked at so far.  Blackburn is obviously a world-class scholar who has thought very deeply about philosophical issues for his entire career; I can tell I"m going to enjoy reading the book with a sense that he could run philosophical circles around me if he were so inclined.  In contrast to Neuromancer, Snow Crash is light, breezy, but no less profound for that; the world Stephenson paints is anarcho-capitalist and absurd.  The main character, named Hiro Protagonist, is a combination hacker and pizza delivery driver -- well, he loses that job.  Stephenson is having fun at the expense of the dark writing one finds in Neuromancer; this should make this book, though longer, far more readable. Simon  Blackburn

Cyberpunk: Neuromancer -- ends with a whimper

     The real meat of this book is toward the beginning: the tone he sets, the dark world he describes, the Raymond Chandler language, cyberspace, superintelligent AIs.  As for the rest, it is, I suppose, original for its time(? Bladerunner came out just before this novel, one also thinks of movies like Tron), but not that interesting these days; and the characters are flat, very flat, but the beginning is so good that it makes up for the rest .  Eventually Wintermute wins and merges with Neuromancer, creating a more complete intelligence.  We know that Wintermute affects that outside world and Neuromancer is the personality.  The new AI is also apparently in contact with another AI from Alpha Centauri -- maybe they'll get a room. Next time I will take on Snow Crash by Neal Stephenson. I will alternate these posts with a chapter-by-chapter look at Simon Blackburn's Truth: A Guide.

Cyberpunk: Neuromancer -- God

"I AM THAT I AM" -- Exodus Case finally starts getting mad that Wintermute appears to him in the guise of people from his past in Chiba City, to which Wintermute replies: "You want I should come to you in the matrix like a burning bush?"(N, pg. 152)  Wintermute is able to control the reality both inside and outside of cyberspace with what appears to be impunity.   Humans experience Wintermute as a disembodied entity, able to take on many forms, able to speak as though from nowhere, for whom the world is often a plastic plaything.  Now,  this isn't to say that Wintermute and Neuromancer are all-powerful and all-knowing; they aren't, but they have a lot of godlike powers.  It is precisely the fear that they will become too powerful that gives rise to "Turing Police": these are the forces that try to keep artificial intelligences from gaining too much intelligence, which would precipitate, or accelerate the effects of, the technological singularity.

Cyberpunk: Neuromancer -- ROM Constructs and Wintermute

  More is revealed about the nature of the constructs, that is, the electronic copies, of personalities as Case continues to interact with the ROM construct, Dixie McCoy, aka Flatline.  Note Hosaka is a device that can be queried for information, sort of like wikipedia; the Dixie construct looked itself up and discovered he's dead: "How you doing, Dixie?" "I'm dead, Case.  Got enough time on this Hosaka to figure that one." "How's it feel?" "It doesn't." "Bother you?" "What bothers me is, nothin' does." "How's that?" "Had me this buddy in the Russian camp, Siberia, his thumb was frostbit.  Medics came by and cut it off.  Month leater he's tossin' all night/ Elroy, I said, what's eating you?  G**mn thumb's itchin', he says.  So I told him, scratch it.  McCoy, he says, it the other G**mn thumb ... Do me a favor, boy." "What's that, Dix?"

Cyberpunk: Neuromancer -- Artificial Intelligence

     "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)

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 t

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,

Cyberpunk and Post Cyberpunk

     For my next project, I will read 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.

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, 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 i

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 one strange loop constituting the identity of the primary person associated with that brain, but of many strange-loop patterns that are coarse-grained copie

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: &

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

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 propos

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

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 pow