HOUSE_OVERSIGHT_015937.jpg
1.52 MB
Extraction Summary
3
People
1
Organizations
0
Locations
0
Events
0
Relationships
4
Quotes
Document Information
Type:
Book page / scientific article (evidence document)
File Size:
1.52 MB
Summary
This document appears to be a page (p. 247) from a book or scientific paper discussing theoretical computer science and mathematics. It covers Gödel's incompleteness theorem, Turing's proofs regarding algorithms, and the concept of a 'logic limit' in computing. The document is stamped with 'HOUSE_OVERSIGHT_015937', indicating it was produced as evidence during a House Oversight Committee investigation, likely related to Epstein's connections to the scientific community or funding.
People (3)
| Name | Role | Context |
|---|---|---|
| Gödel | Mathematician |
Mentioned in relation to the 'incompleteness theorem' regarding mathematical rules.
|
| Turing | Mathematician/Computer Scientist |
Mentioned as proving that algorithms cannot find solutions for arbitrary problems and that there is no general purpos...
|
| Wiles | Mathematician |
Referenced in the header 'The Wiles Paradox' (likely referring to Andrew Wiles).
|
Organizations (1)
| Name | Type | Context |
|---|---|---|
| House Oversight Committee |
Implied by the Bates stamp 'HOUSE_OVERSIGHT_015937' at the bottom of the page.
|
Key Quotes (4)
"The number of ways to do this is not just infinity, but two to the power of infinity."Source
HOUSE_OVERSIGHT_015937.jpg
Quote #1
"That is Gödel incompleteness theorem."Source
HOUSE_OVERSIGHT_015937.jpg
Quote #2
"Turing showed us there can be no general purpose, mechanical procedure capable of finding solutions to arbitrary problems."Source
HOUSE_OVERSIGHT_015937.jpg
Quote #3
"The solution to this apparent contradiction is to propose a boundary: a ‘logic limit’ above which computers may not solve problems."Source
HOUSE_OVERSIGHT_015937.jpg
Quote #4
Full Extracted Text
Complete text extracted from the document (2,337 characters)
Software 247
denotes a set of sequential steps we can apply to demonstrate this fact.
These steps must have meaning and obey the rules of mathematics, but
what are these rules? Are they written down in a text book?
It turns out there is no way to find this set of rules; it is a super-
infinite task. We would need to reach into our infinite bag of numbers
and pull out rule after rule, turning each into a mathematical model
that explains numbers and logic and what can be done with them to
form mathematical statements. The number of ways to do this is not just
infinity, but two to the power of infinity. This is the number of ways to
permute all possible mathematical rules.
Your mind may be rebelling at this. Surely, if I have an infinite
set of numbers I can just pluck all the numbers from my bag and then
I am certain to have the solution. Unfortunately, it turns out there is
no complete, consistent set of rules; no valid dictionary that maps all
numbers to all of mathematics. That is Gödel incompleteness theorem.
Despite a fundamental limit on mapping all numbers to all of
mathematics, there might still have been an algorithm which could
practically find solutions for a given arbitrary problem. Turing proved
this is not the case.
These steps must have meaning and obey the rules of mathematics, but
what are these rules? Are they written down in a text book?
It turns out there is no way to find this set of rules; it is a super-
infinite task. We would need to reach into our infinite bag of numbers
and pull out rule after rule, turning each into a mathematical model
that explains numbers and logic and what can be done with them to
form mathematical statements. The number of ways to do this is not just
infinity, but two to the power of infinity. This is the number of ways to
permute all possible mathematical rules.
Your mind may be rebelling at this. Surely, if I have an infinite
set of numbers I can just pluck all the numbers from my bag and then
I am certain to have the solution. Unfortunately, it turns out there is
no complete, consistent set of rules; no valid dictionary that maps all
numbers to all of mathematics. That is Gödel incompleteness theorem.
Despite a fundamental limit on mapping all numbers to all of
mathematics, there might still have been an algorithm which could
practically find solutions for a given arbitrary problem. Turing proved
this is not the case.
The Wiles Paradox
Turing showed us there can be no general purpose, mechanical procedure
capable of finding solutions to arbitrary problems. A computer program
cannot discover mathematical theorems nor write programs to do so. Yet
computers regularly solve problems and generate programs. That’s what
software compilers do. This seems to be contradiction.
The solution to this apparent contradiction is to propose a boundary:
a ‘logic limit’ above which computers may not solve problems. With a
high boundary a general-purpose machine could solve most problems
in the real world, though some esoteric mathematical puzzles would be
beyond it. But if the boundary were low, many activities in our daily life
would need some sort of alternative, creative thinking. It is crucial to
know where the logic limit lies.
Turing showed us there can be no general purpose, mechanical procedure
capable of finding solutions to arbitrary problems. A computer program
cannot discover mathematical theorems nor write programs to do so. Yet
computers regularly solve problems and generate programs. That’s what
software compilers do. This seems to be contradiction.
The solution to this apparent contradiction is to propose a boundary:
a ‘logic limit’ above which computers may not solve problems. With a
high boundary a general-purpose machine could solve most problems
in the real world, though some esoteric mathematical puzzles would be
beyond it. But if the boundary were low, many activities in our daily life
would need some sort of alternative, creative thinking. It is crucial to
know where the logic limit lies.
The Logic Limit
Amazingly, in many branches of science it is possible to pinpoint the exact
location of the logic limit, but finding that boundary in mathematics has
taken forty years work from some of the greatest mathematicians of the
20th century.
Amazingly, in many branches of science it is possible to pinpoint the exact
location of the logic limit, but finding that boundary in mathematics has
taken forty years work from some of the greatest mathematicians of the
20th century.
HOUSE_OVERSIGHT_015937
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