HOUSE_OVERSIGHT_015894.jpg
1.6 MB
Extraction Summary
2
People
1
Organizations
0
Locations
1
Events
1
Relationships
4
Quotes
Document Information
Type:
Book excerpt / evidence document
File Size:
1.6 MB
Summary
This document appears to be page 204 from a book titled 'Are the Androids Dreaming Yet?', which has been included in a House Oversight Committee investigation (Bates stamp HOUSE_OVERSIGHT_015894). The text is a philosophical and mathematical discussion regarding Gödel's Incompleteness Theorems, the liar's paradox, and the concept of inconsistency in mathematical models. It details David Hilbert's angry reaction to Gödel's work and discusses the implications of these theorems on human creativity and knowledge discovery.
People (2)
| Name | Role | Context |
|---|---|---|
| Kurt Gödel | Mathematician |
Subject of the text; his incompleteness theorems are discussed.
|
| David Hilbert | Mathematician |
His reaction (anger) to Gödel's proof and subsequent abandonment of formalism is described.
|
Organizations (1)
| Name | Type | Context |
|---|---|---|
| House Oversight Committee |
Inferred from the Bates stamp 'HOUSE_OVERSIGHT' at the bottom of the page.
|
Timeline (1 events)
Historical (Post-Gödel's proof)
Hilbert's reaction to Gödel's proof where he stopped working on formalism.
Unknown
Relationships (1)
Hilbert reacted with anger to Gödel's proof which disproved Hilbert's 30 years of work on mathematical completeness.
Key Quotes (4)
"“This proof number is not a proof of the truth of this theorem number.”"Source
HOUSE_OVERSIGHT_015894.jpg
Quote #1
"“Within any formal system of mathematics there can be statements that are true but are not provable using only the rules of that system.”"Source
HOUSE_OVERSIGHT_015894.jpg
Quote #2
"When Hilbert heard of Gödel’s proof, his first reaction was anger."Source
HOUSE_OVERSIGHT_015894.jpg
Quote #3
"The repercussions of Gödel’s theory have more to do with understanding our place in the Universe and the nature of knowledge discovery."Source
HOUSE_OVERSIGHT_015894.jpg
Quote #4
Full Extracted Text
Complete text extracted from the document (2,428 characters)
204 Are the Androids Dreaming Yet?
Every correctly formed theorem number has another number, which demonstrates the proof of that number.
If this is universally true there should be no contradiction. Unfortunately if you apply the theorem to itself you get something similar to the liar’s paradox.
“This proof number is not a proof of the truth of this theorem number.”
The proof number proves the theorem number is true, but the truth of the statement is that it can’t be a proof of the statement… Paradox.
The only way to resolve the paradox is to go back one step and realize that not every correctly formed theorem number has a proof number using only the rules of that system.
Concisely, Gödel’s theorem says, “Within any formal system of mathematics there can be statements that are true but are not provable using only the rules of that system.”
When Hilbert heard of Gödel’s proof, his first reaction was anger. After all, he had spent 30 years of his life trying to prove mathematics was tidy and complete. Gödel had just shown it was not. Hilbert never worked on formalism again, but the rest of the mathematical establishment largely ignored the result. Gödel’s proof did not stop mathematicians proving new theorems nor doing useful mathematics. They went on much as before, using a mixture of intuition and analysis. The only difference was someone had told them analysis alone would not succeed. The repercussions of Gödel’s theory have more to do with understanding our place in the Universe and the nature of knowledge discovery. These are ‘big’ philosophical questions, which don’t greatly affect the day-to-day ability of a mathematician to do their job. However, it is important to understand that knowledge discovery is not simply analysis. Knowing this helps us understand human creativity.
If this is universally true there should be no contradiction. Unfortunately if you apply the theorem to itself you get something similar to the liar’s paradox.
“This proof number is not a proof of the truth of this theorem number.”
The proof number proves the theorem number is true, but the truth of the statement is that it can’t be a proof of the statement… Paradox.
The only way to resolve the paradox is to go back one step and realize that not every correctly formed theorem number has a proof number using only the rules of that system.
Concisely, Gödel’s theorem says, “Within any formal system of mathematics there can be statements that are true but are not provable using only the rules of that system.”
When Hilbert heard of Gödel’s proof, his first reaction was anger. After all, he had spent 30 years of his life trying to prove mathematics was tidy and complete. Gödel had just shown it was not. Hilbert never worked on formalism again, but the rest of the mathematical establishment largely ignored the result. Gödel’s proof did not stop mathematicians proving new theorems nor doing useful mathematics. They went on much as before, using a mixture of intuition and analysis. The only difference was someone had told them analysis alone would not succeed. The repercussions of Gödel’s theory have more to do with understanding our place in the Universe and the nature of knowledge discovery. These are ‘big’ philosophical questions, which don’t greatly affect the day-to-day ability of a mathematician to do their job. However, it is important to understand that knowledge discovery is not simply analysis. Knowing this helps us understand human creativity.
Inconsistency
In the proof above, I said the only way to resolve the paradox is by saying there cannot be a proof number for every mathematical statement and therefore mathematics is incomplete. There is one other way to solve the paradox, and that is by allowing inconsistency into the system. Gödel’s proof assumes you can prove something true or false, but what if you could prove it true and false? In this case, the system is complete but you can prove truths and untruths within it! This may seem an acceptable solution, but inconsistency in a mathematical model is a cancer that will
In the proof above, I said the only way to resolve the paradox is by saying there cannot be a proof number for every mathematical statement and therefore mathematics is incomplete. There is one other way to solve the paradox, and that is by allowing inconsistency into the system. Gödel’s proof assumes you can prove something true or false, but what if you could prove it true and false? In this case, the system is complete but you can prove truths and untruths within it! This may seem an acceptable solution, but inconsistency in a mathematical model is a cancer that will
HOUSE_OVERSIGHT_015894
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