Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
repetitive behavior very quickly. Or, to take a silly example, a car will never pick up the idea, no matter how much or how well ...
boring ritual of checkmating. Although it lost every game it played, it did it in style. A lot of local chess experts were impre ...
also have some figuring on it. We will occasionally refer back to these two modes of dealing with a formal system, and we will c ...
Step 3: Apply every applicable rule to the theorems produced in step 2. This yields five new theorems: MIIIIU, MIIUIIU, MIUIUIUl ...
amount of time, then that test is called a decision procedure for the given formal system. When you have a decision procedure, t ...
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Two-Part! nvention or, What the Tortoise Said to Achilles by Lewis CarrolP Achilles had overtaken the Tortoise, and had seated h ...
Schools are invented, which will not be till some two thousand years later-will grant THAT." "And if some reader had NOT yet acc ...
"And why must I?" "Because it follows LOGICALLY from them. If A and Band C are true, Z MUST be true. You can't dispute THAT, I i ...
CHAPTER II Meaning and Form in Mathematics THIS Two-Part Invention was the inspiration for my two characters. Just as Lewis Carr ...
DEFINITION: X P - q X - is an axiom, whenever X is composed of hyphens only. Note that 'x' must stand for the same string of hyp ...
hyphen-group. For instance, --p--q----is a theorem, since 2 plus 2 equals 4, whereas --p--q-is not, since 2 plus 2 is not 1. To ...
(la) Throw the simplest possible aXIOm (-p-q--) into the bucket. (1 b) Apply the rule of inference to the item in the bucket, an ...
It is cause for joy when a mathematician discovers an isomorphism between two structures which he knows. It is often a "bolt fro ...
theorems reflect some portion of reality isomorphically. In such a case, the choice of symbols is a highly motivated one, as is ...
in a language, when we have learned a meaning for a word, we then make new statements based on the meaning of the word. In a sen ...
tion will be meaningful to the extent that it accurately reflects some iso- morphism to the real world. When different aspects o ...
system. Its symbols do not move around on paper, but rather in a three- dimensional vacuum (space); they are the elementary part ...
any time in their lives drawn a 12 by 12 rectangle, and then counted the little squares in it? Most people would regard the draw ...
many cubes assembled to form a large rectangular solid. Multiplicative commutativity and associativity are just the assumptions ...
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