"Godel's Theorem has many profound implications, both for science and for philosophy. ... Godel's message is that mankind will never know the final secret of the universe by 'finitistic' or constructivistic thought alone; it's impossible for human beings ever to formulate a complete description of the natural numbers. There will always be arithmetic truths that escape our ability to fence them in by any kind of finite analysis. As Rudy Rucker has expressed it, Godel's Theorem leaves scientists in a position similar to that of Joseph K. in Kafka's novel 'The Trial'. We scurry around, running up and down endless corridors, buttonholing people, going in and out of offices, and, in general, conducting investigations. But we will never achieve ultimate success; there is no final verdict in the court of science leading to absolute truth. However, Rucker notes, "To understand the labyrinthine nature of the castle [i.e., court] is, somehow, to be free of it." And there's no understanding of the court of science that digs deeper into its foundations that the understanding given by Godel's Theorem."
Casti, John - Searching for Certainty
skica dokaza:
The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:
1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all. 2. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine. 3. Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true." 4. Now Gödel laughs his high laugh and asks UTM whether G is true or not. 5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements. 6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true"). 7. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."
Think about it - it grows on you ...
With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics ...
Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.
Rucker, Infinity and the mind
20.03.2007. (14:43)
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Gödelov teorem u biti glasi - Sustav ne može u isto vrijeme biti konzistnentan i konačan, odnosno: ono nedovršeno i ono dinamično je Bolje od onog statičnog i onog konačnog.
Robert Pirsig u svojim dvijema knjigama (Zen and the Art of Motorcycle Maintenance: An Inquiry into Values (1974) i Lila: An Inquiry into Morals (1991) ) koncipira zametnutu ideju kvalitete (što je dobro Fedro?) ističući upravo kvalitetu kao - to proton kinoun -- primum movens. I tako povezao Dobro sa Dinamičnim.
sa wiki:
Quality
"Quality", or "value" as described by Pirsig, cannot be defined because it empirically precedes any intellectual constructions. It is the "knife-edge" of experience, known to all. "What distinguishes good and bad writing? Do we need to ask this question of Lysias or anyone else who ever did write anything?" (Plato's Phaedrus, 258d). Likening it with the Tao, Pirsig believes that Quality is the fundamental force in the universe stimulating everything from atoms to animals to evolve and incorporate ever greater levels of Quality. According to the MOQ, everything (including mind, ideas and matter) is a product and a result of Quality.
Static and Dynamic qualities
The MOQ(Metaphysics Of Quality) divides Quality into two forms: static quality patterns (patterned) and Dynamic Quality (unpatterned). The four patterns of static value as well as Dynamic Quality account exhaustively for all of reality. As the initial (cutting edge) Dynamic Quality become habituated, it turns into static patterns. It is important to note that Pirsig is not proposing a duality: Quality is one, yet manifests itself differently.
Dynamic Quality
Dynamic Quality includes everything not static. Dynamic Quality is the force of change in the universe; when this aspect of Quality becomes habitual or customary it becomes static. Pirsig called Dynamic Quality "the pre-intellectual cutting edge of reality" because it can be recognized before one can think about it. What does it mean ? Quality lies in the moment we sense anything during the instantaneous present; with a short delay we give this impression a static form by describing it as emotion, thing, word etc. These static forms, if they have enough good or bad quality, are given names and are interchanged with other people, building the base of knowledge for a culture. So some cultures divide between things other cultures perceive as equal (Pirsig give as example the sounds of the Indian syllables "dha" and "da" which are absolutely equal for western ears) and some cultures haven't any words for a specific meaning at all (The exact meaning of the German word "verklemmt" cannot be translated in English). It also explains that the Dynamic beauty of a piece of music can be recognized before reading any static analysis explaining why the music is beautiful.
Platonova (jedna od formi) ideja Dobra (koja je istovjetna ideji kršćanskog Boga prema nekim ) je u smislu prožetosti, napetosti : "My opinion is that in the world of knowledge the idea of good (the Good) appears last of all, and is seen only with an effort; and, when seen, is also inferred to be the universal author of all things beautiful and right, parent of light and of the lord of light in this visible world, and the immediate source of reason and truth in the intellectual." (517b,c) Republic Book 7
i neizostavni Bela:
Bela Hamvas u Scientia Sacra II - Kršćanstvo :
Božje prisustvo nije u gorkom i divljem mraku prve prirode. Ali nije ni u izjednačenju, ni u miru, ni u svjetlosti. Bog je u transformaciji, u kojoj mahnitost hvata za gušu sila koja je jača od nje. Bog je u kretnji iz koje zrači i koja se u samoprijegoru okreće protiv vlastite prirode, slavodobitno izlazi iz tog zaokreta, istodobno stoji u opreci s mračnim silama, ponovno se okreće protiv sebe i ponovno svladava sebe. Napor da stalno bude on sam, istodobno da bi bio iznad sebe,da živi i da pobijedi sebe. Da razriješi sve svoje strasti i da ih drži na uzdi. Da potone u mraku i da se pokaže kao svjetlost.
A na istoku Shiva pleše već tisućama godina...
20.03.2007. (15:40)
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Luce
Nemoj ti meni o nepotpunosti matematike! Popizdim!
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seismic riffs
in the meantime...whiskeybar.blog.hr..
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NEMANJA
Nisam ja, majkemi, učiteljice, Kurt je!
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Luce
Kažem ja, ništa kurtoazno nije dobro.
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NEMANJA
Znao sam ja i prije da matematika nije potpuna. Kad bolje pogledaš te brojeve, zar nemaš dojam da tu nekaj fali? Ne znam točno koji, ali jedan fali!
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Luce
Meni uvijek jedan fali, taj jedan, gad jedan.
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NEMANJA
Koliko god puta da brojim uvijek mi ispadne drukčiji rezultat!
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Luce
Ja uvijek isto prebrojim, to je dobra vijest.
I uvijek mi nešto fali, to je malo lošije.
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vertebrata
nemanja - primjeti Das Schloss : ;)
"Godel's Theorem has many profound implications, both for science and for philosophy. ... Godel's message is that mankind will never know the final secret of the universe by 'finitistic' or constructivistic thought alone; it's impossible for human beings ever to formulate a complete description of the natural numbers. There will always be arithmetic truths that escape our ability to fence them in by any kind of finite analysis. As Rudy Rucker has expressed it, Godel's Theorem leaves scientists in a position similar to that of Joseph K. in Kafka's novel 'The Trial'. We scurry around, running up and down endless corridors, buttonholing people, going in and out of offices, and, in general, conducting investigations. But we will never achieve ultimate success; there is no final verdict in the court of science leading to absolute truth. However, Rucker notes, "To understand the labyrinthine nature of the castle [i.e., court] is, somehow, to be free of it." And there's no understanding of the court of science that digs deeper into its foundations that the understanding given by Godel's Theorem."
Casti, John - Searching for Certainty
skica dokaza:
The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:
1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
2. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
3. Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
4. Now Gödel laughs his high laugh and asks UTM whether G is true or not.
5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
7. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."
Think about it - it grows on you ...
With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics ...
Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.
Rucker, Infinity and the mind
20.03.2007. (14:43) - - - - - promjene spremljene- uredi komentar - obriši komentar - prijavi ovaj komentar kao spam - zabrani komentiranje autoru ovog komentara- učitavam...
vertebrata
Od Gödela do Boga ;)
Malo analogija:
Gödelov teorem u biti glasi - Sustav ne može u isto vrijeme biti konzistnentan i konačan, odnosno: ono nedovršeno i ono dinamično je Bolje od onog statičnog i onog konačnog.
Robert Pirsig u svojim dvijema knjigama (Zen and the Art of Motorcycle Maintenance: An Inquiry into Values (1974) i Lila: An Inquiry into Morals (1991) ) koncipira zametnutu ideju kvalitete (što je dobro Fedro?) ističući upravo kvalitetu kao - to proton kinoun -- primum movens. I tako povezao Dobro sa Dinamičnim.
sa wiki:
Quality
"Quality", or "value" as described by Pirsig, cannot be defined because it empirically precedes any intellectual constructions. It is the "knife-edge" of experience, known to all. "What distinguishes good and bad writing? Do we need to ask this question of Lysias or anyone else who ever did write anything?" (Plato's Phaedrus, 258d). Likening it with the Tao, Pirsig believes that Quality is the fundamental force in the universe stimulating everything from atoms to animals to evolve and incorporate ever greater levels of Quality. According to the MOQ, everything (including mind, ideas and matter) is a product and a result of Quality.
Static and Dynamic qualities
The MOQ(Metaphysics Of Quality) divides Quality into two forms: static quality patterns (patterned) and Dynamic Quality (unpatterned). The four patterns of static value as well as Dynamic Quality account exhaustively for all of reality. As the initial (cutting edge) Dynamic Quality become habituated, it turns into static patterns. It is important to note that Pirsig is not proposing a duality: Quality is one, yet manifests itself differently.
Dynamic Quality
Dynamic Quality includes everything not static. Dynamic Quality is the force of change in the universe; when this aspect of Quality becomes habitual or customary it becomes static. Pirsig called Dynamic Quality "the pre-intellectual cutting edge of reality" because it can be recognized before one can think about it. What does it mean ? Quality lies in the moment we sense anything during the instantaneous present; with a short delay we give this impression a static form by describing it as emotion, thing, word etc. These static forms, if they have enough good or bad quality, are given names and are interchanged with other people, building the base of knowledge for a culture. So some cultures divide between things other cultures perceive as equal (Pirsig give as example the sounds of the Indian syllables "dha" and "da" which are absolutely equal for western ears) and some cultures haven't any words for a specific meaning at all (The exact meaning of the German word "verklemmt" cannot be translated in English). It also explains that the Dynamic beauty of a piece of music can be recognized before reading any static analysis explaining why the music is beautiful.
Platonova (jedna od formi) ideja Dobra (koja je istovjetna ideji kršćanskog Boga prema nekim ) je u smislu prožetosti, napetosti : "My opinion is that in the world of knowledge the idea of good (the Good) appears last of all, and is seen only with an effort; and, when seen, is also inferred to be the universal author of all things beautiful and right, parent of light and of the lord of light in this visible world, and the immediate source of reason and truth in the intellectual." (517b,c) Republic Book 7
i neizostavni Bela:
Bela Hamvas u Scientia Sacra II - Kršćanstvo :
Božje prisustvo nije u gorkom i divljem mraku prve prirode. Ali nije ni u izjednačenju, ni u miru, ni u svjetlosti. Bog je u transformaciji, u kojoj mahnitost hvata za gušu sila koja je jača od nje. Bog je u kretnji iz koje zrači i koja se u samoprijegoru okreće protiv vlastite prirode, slavodobitno izlazi iz tog zaokreta, istodobno stoji u opreci s mračnim silama, ponovno se okreće protiv sebe i ponovno svladava sebe. Napor da stalno bude on sam, istodobno da bi bio iznad sebe,da živi i da pobijedi sebe. Da razriješi sve svoje strasti i da ih drži na uzdi. Da potone u mraku i da se pokaže kao svjetlost.
A na istoku Shiva pleše već tisućama godina...
20.03.2007. (15:40) - - - - - promjene spremljene- uredi komentar - obriši komentar - prijavi ovaj komentar kao spam - zabrani komentiranje autoru ovog komentara- učitavam...
Diabolis
samo pohvale za tu knjigu
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