Editing 2610: Assigning Numbers
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==Explanation== | ==Explanation== | ||
+ | {{incomplete|Created by YÖDEL'S COMPLETENESS THEOREM - Please change this comment when editing this page. Do NOT delete this tag too soon.}} | ||
'''This explanation is by mathematical necessity either incomplete or incorrect.''' | '''This explanation is by mathematical necessity either incomplete or incorrect.''' | ||
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So, as well as being the mechanism that underlies one of the most profound theorems of 20th century mathematics, it can be mis-used for all kinds of bad or misguided science. From Cueball's attitude, it is far from clear that his attempt will reliably translate his project into a numerical system, nor that his attempt to "do math on it!" will be any more competent. | So, as well as being the mechanism that underlies one of the most profound theorems of 20th century mathematics, it can be mis-used for all kinds of bad or misguided science. From Cueball's attitude, it is far from clear that his attempt will reliably translate his project into a numerical system, nor that his attempt to "do math on it!" will be any more competent. | ||
− | One of the major characters who looked at the concept is Kurt Gödel. He introduced the idea of {{w|Gödel numbering}} with his landmark {{w|incompleteness theorems}}. In it a unique natural number is assigned to each | + | One of the major characters who looked at the concept is Kurt Gödel. He introduced the idea of {{w|Gödel numbering}} with his landmark {{w|incompleteness theorems}}. In it a unique natural number is assigned to each atom, statement, and proof, which might otherwise be difficult to accurately process in any other kind of approach. Instead, it is now possible to create metamathematical statements in the language of mathematics. |
− | + | As the formulation in use was itself definable with numbers, in doing so, Gödel also proved the existence of a statement that essentially said "this statement has no proof." If the statement could be proven, the statement would there be false, and the mathematics being used would be inconsistent. Gödel presumed that the only other possibility was that the statement is true without a mathematical proof, and therefore the mathematics of the system is incomplete; however, since the statement is self referencing, a third possibility exists: The statement might actually be paradoxical. (This third possibility is presented here http://dstoner.net/Math_Science/godel.html.) Gödel's theorem led to a fundamental reckoning in the world of mathematics when it was published. | |
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The title text suggests that Gödel should perform such an analysis on different branches of mathematics, by calculating the average of all the fields' theorems' Gödel numbers. This is nonsensical for a number of reasons: | The title text suggests that Gödel should perform such an analysis on different branches of mathematics, by calculating the average of all the fields' theorems' Gödel numbers. This is nonsensical for a number of reasons: | ||
− | :1) Gödel is long dead, and dead people can't write articles; | + | :1) Gödel is long dead, and dead people can't write articles; |
:2) Gödel numbers grow very large very quickly, and depend heavily on the specific values assigned to each logical operator. Therefore the results could be manipulated simply by changing the numbering order of each operator; | :2) Gödel numbers grow very large very quickly, and depend heavily on the specific values assigned to each logical operator. Therefore the results could be manipulated simply by changing the numbering order of each operator; | ||
− | :3) It may be very hard to gather all theorems in a field, or even a representative sample | + | :3) It may be very hard to gather all theorems in a field, or even a representative sample; |
− | If anyone were to attempt this form of analysis, it would be an example of the bad data science described in the | + | :4) Different fields of science, like biology or human behaviour, may not be able to write their theorems in the mathematical language of Gödel's incompleteness theorem |
+ | If anyone were to attempt this form of analysis, it would be an example of the bad data science described in the title text. | ||
==Transcript== | ==Transcript== | ||
− | :[Cueball | + | :[Cueball hold a hand up to his chin while he ponders the contents of what may be a whiteboard. There are five general lines of unreadable scribbling on the board, and between the two bottom lines, there is a square frame to the right with another scribble to the left. Cueball's thoughts are shown above him in a large thought bubble.] |
:Cueball's thinking: If I assign numbers to each of these things, then it becomes '''''data''''', and I can do '''''math''''' on it! | :Cueball's thinking: If I assign numbers to each of these things, then it becomes '''''data''''', and I can do '''''math''''' on it! | ||
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{{comic discussion}} | {{comic discussion}} | ||
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