Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter
Authors: Douglas R. Hofstadter
Tags: General, science, Biography & Autobiography, music, Computers, Artificial intelligence, Genres & Styles, Philosophy, Art, Science & Technology, Mathematics, Individual Artists, Classical, Logic, Symmetry, Bach; Johann Sebastian, Metamathematics, Intelligence (AI) & Semantics, G'odel; Kurt, Escher; M. C
Ads: Link
interpretations.
Active vs. Passive Meanings
Probably the most significant fact of this Chapter, if understood deeply this: the pqsystem seems to force us into recognizing that symbols of a formal system, though initially without meaning, cannot avoid taking on "meaning" of sorts at least if an isomorphism is found. The difference between meaning it formal system and in a language is a very important one, however. It is this:
in a language, when we have learned a meaning for a word, we then mar-c new statements based on the meaning of the word. In a sense the meaning becomes active , since it brings into being a new rule for creating sentences. This means that our command of language is not like a finished product: the rules for making sentences increase when we learn new meanings. On the other hand, in a formal system, the theorems are predefined, by the rules of production. We can choose "meanings" based on an isomorphism (if we can find one) between theorems and true statements. But this does not give us the license to go out and add new theorems to the established theorems. That is what the Requirement of Formality in Chapter I was warning you of.
In the MIU-system, of course, there was no temptation to go beyond the four rules, because no interpretation was sought or found. But here, in our new system, one might be seduced by the newly found "meaning" of each symbol into thinking that the string
--p--p--p--q
is a theorem. At least, one might wish that this string were a theorem. But wishing doesn't change the fact that it isn't. And it would be a serious mistake to think that it "must" be a theorem, just because 2 plus 2 plus 2 plus 2 equals 8. It would even be misleading to attribute it any meaning at all, since it is not well-formed, and our meaningful interpretation is entirely derived from looking at well-formed strings.
In a formal system, the meaning must remain passive ; we can read each string according to the meanings of its constituent symbols, but we do not have the right to create new theorems purely on the basis of the meanings we've assigned the symbols.
Interpreted formal systems straddle the line between systems without meaning, and systems with meaning. Their strings can be thought of as "expressing" things, but this must come only as a consequence of the formal properties of the system.
Double-Entendre!
And now, I want to destroy any illusion about having found the meanings for the symbols of the pq-system. Consider the following association:
p <= => equals
q <= => taken from
- <= => one
-- <= => two
etc.
Now, --p---q---- has a new interpretation: "2 equals 3 taken from 5". Of course it is a true statement. All theorems will come out true under this new interpretation. It is just as meaningful as the old one. Obviously, it is silly to ask, "But which one is the meaning of the string?" An interpreta
tion will me meaningful to the extent that it accurately reflects some isomorphism to the real world. When different aspects of the real world a isomorphic to each other (in this case, additions and subtractions), or single formal system can be isomorphic to both, and therefore can take ( two passive meanings. This kind of double-valuedness of symbols at strings is an extremely important phenomenon. Here it seems trivial curious, annoying.
But it will come back in deeper contexts and bring with it a great richness of ideas.
Here is a summary of our observations about the pq-system. Und either of the two meaningful interpretations given, every well-form( string has a grammatical assertion for its counterpart-some are true, son false. The idea of well formed strings in any formal system is that they a those strings which, when interpreted symbol for symbol, yield grammatical sentences. (Of course, it depends on the interpretation, but usually, there one in mind.) Among the well-formed strings occur the theorems. The: are defined by an axiom schema, and a rule of production. My goal
Active vs. Passive Meanings
Probably the most significant fact of this Chapter, if understood deeply this: the pqsystem seems to force us into recognizing that symbols of a formal system, though initially without meaning, cannot avoid taking on "meaning" of sorts at least if an isomorphism is found. The difference between meaning it formal system and in a language is a very important one, however. It is this:
in a language, when we have learned a meaning for a word, we then mar-c new statements based on the meaning of the word. In a sense the meaning becomes active , since it brings into being a new rule for creating sentences. This means that our command of language is not like a finished product: the rules for making sentences increase when we learn new meanings. On the other hand, in a formal system, the theorems are predefined, by the rules of production. We can choose "meanings" based on an isomorphism (if we can find one) between theorems and true statements. But this does not give us the license to go out and add new theorems to the established theorems. That is what the Requirement of Formality in Chapter I was warning you of.
In the MIU-system, of course, there was no temptation to go beyond the four rules, because no interpretation was sought or found. But here, in our new system, one might be seduced by the newly found "meaning" of each symbol into thinking that the string
--p--p--p--q
is a theorem. At least, one might wish that this string were a theorem. But wishing doesn't change the fact that it isn't. And it would be a serious mistake to think that it "must" be a theorem, just because 2 plus 2 plus 2 plus 2 equals 8. It would even be misleading to attribute it any meaning at all, since it is not well-formed, and our meaningful interpretation is entirely derived from looking at well-formed strings.
In a formal system, the meaning must remain passive ; we can read each string according to the meanings of its constituent symbols, but we do not have the right to create new theorems purely on the basis of the meanings we've assigned the symbols.
Interpreted formal systems straddle the line between systems without meaning, and systems with meaning. Their strings can be thought of as "expressing" things, but this must come only as a consequence of the formal properties of the system.
Double-Entendre!
And now, I want to destroy any illusion about having found the meanings for the symbols of the pq-system. Consider the following association:
p <= => equals
q <= => taken from
- <= => one
-- <= => two
etc.
Now, --p---q---- has a new interpretation: "2 equals 3 taken from 5". Of course it is a true statement. All theorems will come out true under this new interpretation. It is just as meaningful as the old one. Obviously, it is silly to ask, "But which one is the meaning of the string?" An interpreta
tion will me meaningful to the extent that it accurately reflects some isomorphism to the real world. When different aspects of the real world a isomorphic to each other (in this case, additions and subtractions), or single formal system can be isomorphic to both, and therefore can take ( two passive meanings. This kind of double-valuedness of symbols at strings is an extremely important phenomenon. Here it seems trivial curious, annoying.
But it will come back in deeper contexts and bring with it a great richness of ideas.
Here is a summary of our observations about the pq-system. Und either of the two meaningful interpretations given, every well-form( string has a grammatical assertion for its counterpart-some are true, son false. The idea of well formed strings in any formal system is that they a those strings which, when interpreted symbol for symbol, yield grammatical sentences. (Of course, it depends on the interpretation, but usually, there one in mind.) Among the well-formed strings occur the theorems. The: are defined by an axiom schema, and a rule of production. My goal
Similar Books
