Gödel, Escher, Bach

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  第94页

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  A full formalization of geometry would take the drastic step of making _every_ term undefined---that is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because ... the symbols automatically pick up passive meanings in accordance with the theorems they occur in. It is another question, though, whether people discover those me... (更多)

  A full formalization of geometry would take the drastic step of making _every_ term undefined---that is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because ... the symbols automatically pick up passive meanings in accordance with the theorems they occur in. It is another question, though, whether people discover those meanings, for to do so requires finding a set of concepts which can be linked by an isomorphism to the symbols in the formal system. If one begins with the aim of formalizing geometry, presumably one has an _intended_ interpretation for each symbol, so that the passive meanings are built into the system.

  consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system.

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  2011-12-04 09:35:44   回应

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  第79页

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  Tortoise: ... By the way, did you know that each of the four letters in Bach's name is the name of a musical note? Achilles: 'tisn't possible, is it? After all, musical notes only go from 'A' through 'G'. Tortoise: Just so; in most countries, that's the case. But in Germany, Bach's own homeland, the convention has always been similar, except that what we call 'B', they call 'H', and what w... (更多)

  Tortoise: ... By the way, did you know that each of the four letters in Bach's name is the name of a musical note? Achilles: 'tisn't possible, is it? After all, musical notes only go from 'A' through 'G'. Tortoise: Just so; in most countries, that's the case. But in Germany, Bach's own homeland, the convention has always been similar, except that what we call 'B', they call 'H', and what we call 'B-flat', they call 'B'. For instance, we talk about Bach's "Mass in B Minor", whereas they talk about his "H-moll Messe".

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  2011-12-01 07:28:07   回应

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  第78页

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  By elaborate calculations, its little computer figured out what effects the sounds would have upon its phonograph. If it deduced that the sounds were such that they would cause the machine in its present configuration to break, then it did something very clever. Old omega contained a device which could disassemble large parts of its phonograph subunit, and rebuild them in new ways, so that it coul... (更多)

  By elaborate calculations, its little computer figured out what effects the sounds would have upon its phonograph. If it deduced that the sounds were such that they would cause the machine in its present configuration to break, then it did something very clever. Old omega contained a device which could disassemble large parts of its phonograph subunit, and rebuild them in new ways, so that it could, in effect, change its own structure. If the sounds were "dangerous", a new configuration was chosen, one to which the sounds would pose no threat, and this new configuration would then be built by the rebuilding subunit, under direction of the little computer. Only after this rebuilding operation would Record Player Omega attempt to play the record.

  This is like switching to a new gauge to avoid multi-valuedness in the wavefunction phase. Now, we know the flux can never be avioded simply by gauge transformations... (收起)

  2011-12-01 07:14:05   回应

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  第10页

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  What makes this canon different from any other, however, is that when it concludes---or, rather, seems to conclude---it is no longer in the key of C minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys) right under the listener's nose. And it is so constructed that this "ending" ties smoothly onto the beginning again; thus one can repeat the process and retur... (更多)

  What makes this canon different from any other, however, is that when it concludes---or, rather, seems to conclude---it is no longer in the key of C minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys) right under the listener's nose. And it is so constructed that this "ending" ties smoothly onto the beginning again; thus one can repeat the process and return in the key of E, only to join again to the beginning.

  That's a Wannier flow in the Hofstadter problem right there. (收起)

  2011-11-20 10:33:04   回应

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  第15页

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  Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way? (更多)

  Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way?

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  2011-11-20 10:49:27   回应

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  第18页

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  Whereas the Epimenides statement creates a paradox since it is neither true nor false, the Godel sentence G is unprovable (inside Principia Mathematica) but true. The grand conclusin? That the system of Principia Mathematica is "incomplete"---there are true statements of number theory which its methods of proof are too weak to demonstrate. In short, Godel showed that provability is a ... (更多)

  Whereas the Epimenides statement creates a paradox since it is neither true nor false, the Godel sentence G is unprovable (inside Principia Mathematica) but true. The grand conclusin? That the system of Principia Mathematica is "incomplete"---there are true statements of number theory which its methods of proof are too weak to demonstrate.

  In short, Godel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved.

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  2011-11-20 11:01:14   回应

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  第94页

  zsh

  A full formalization of geometry would take the drastic step of making _every_ term undefined---that is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because ... the symbols automatically pick up passive meanings in accordance with the theorems they occur in. It is another question, though, whether people discover those me... (更多)

  A full formalization of geometry would take the drastic step of making _every_ term undefined---that is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because ... the symbols automatically pick up passive meanings in accordance with the theorems they occur in. It is another question, though, whether people discover those meanings, for to do so requires finding a set of concepts which can be linked by an isomorphism to the symbols in the formal system. If one begins with the aim of formalizing geometry, presumably one has an _intended_ interpretation for each symbol, so that the passive meanings are built into the system.

  consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system.

  (收起)

  2011-12-04 09:35:44   回应

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• 

  第79页

  zsh

  Tortoise: ... By the way, did you know that each of the four letters in Bach's name is the name of a musical note? Achilles: 'tisn't possible, is it? After all, musical notes only go from 'A' through 'G'. Tortoise: Just so; in most countries, that's the case. But in Germany, Bach's own homeland, the convention has always been similar, except that what we call 'B', they call 'H', and what w... (更多)

  Tortoise: ... By the way, did you know that each of the four letters in Bach's name is the name of a musical note? Achilles: 'tisn't possible, is it? After all, musical notes only go from 'A' through 'G'. Tortoise: Just so; in most countries, that's the case. But in Germany, Bach's own homeland, the convention has always been similar, except that what we call 'B', they call 'H', and what we call 'B-flat', they call 'B'. For instance, we talk about Bach's "Mass in B Minor", whereas they talk about his "H-moll Messe".

  (收起)

  2011-12-01 07:28:07   回应

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• 

  第78页

  zsh

  By elaborate calculations, its little computer figured out what effects the sounds would have upon its phonograph. If it deduced that the sounds were such that they would cause the machine in its present configuration to break, then it did something very clever. Old omega contained a device which could disassemble large parts of its phonograph subunit, and rebuild them in new ways, so that it coul... (更多)

  By elaborate calculations, its little computer figured out what effects the sounds would have upon its phonograph. If it deduced that the sounds were such that they would cause the machine in its present configuration to break, then it did something very clever. Old omega contained a device which could disassemble large parts of its phonograph subunit, and rebuild them in new ways, so that it could, in effect, change its own structure. If the sounds were "dangerous", a new configuration was chosen, one to which the sounds would pose no threat, and this new configuration would then be built by the rebuilding subunit, under direction of the little computer. Only after this rebuilding operation would Record Player Omega attempt to play the record.

  This is like switching to a new gauge to avoid multi-valuedness in the wavefunction phase. Now, we know the flux can never be avioded simply by gauge transformations... (收起)

  2011-12-01 07:14:05   回应

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