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Mathematical thought can't be fully formalised
SupportiveArgument
1
#1189
Gödel’s theorem shows that human creativity can't be fully formalised. The ingenuity of mathematicians in devising new methods can't be reduced to a precise logical form.
For example, it has been shown that humans using "informal" mathematical reasoning, can prove theorems that can't be proven by any formal means.
Ernest Nagel and James R. Newman (1958).
CONTEXT
(Help)
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Artificial Intelligence »
Artificial Intelligence
Artificial Intelligence☜A collaboratively editable version of Robert Horns brilliant and pioneering debate map Can Computers Think?—exploring 50 years of philosophical argument about the possibility of computer thought.☜F1CEB7
▲
Are thinking computers mathematically possible? [7] »
Are thinking computers mathematically possible? [7]
Are thinking computers mathematically possible? [7]☜Is it mathematically possible for a computer to think as well as a human can? Does the mathematics of computation contain anything to prohibit machines from thinking?☜FFB597
▲
No: computers are limited by Gödel's theorems »
No: computers are limited by Gödel's theorems
No: computers are limited by Gödel's theorems☜Gödels theorem proves that a computer cant in principle operate with human understanding (see detailed text). Gödels incompleteness theorems are the Achilles heel of mechanism. John Lucas (1961).☜59C6EF
▲
Theorems show limitations of machine thought »
Theorems show limitations of machine thought
Theorems show limitations of machine thought☜Gödel’s theorem, and other mathematical theorems like it, reveal essential limitations on the project of making machines that think.☜98CE71
■
Mathematical thought can't be fully formalised
Mathematical thought can't be fully formalised☜Gödel’s theorem shows that human creativity cant be fully formalised. The ingenuity of mathematicians in devising new methods cant be reduced to a precise logical form.☜98CE71
●
Gödelian arguments don't affect open proof systems »
Gödelian arguments don't affect open proof systems
Gödelian arguments don't affect open proof systems☜Gödels theorem distinguishes between open and closed proof systems. The former interact with the environment via a stream of inputs, are potentially noncomputable, immune to Gödelian arguments, and may yet be as creative and insightful as humans.☜EF597B
●
Proof of human superiority relies on proof of consistency »
Proof of human superiority relies on proof of consistency
Proof of human superiority relies on proof of consistency☜Newman and Nagels thesis results from a misapplication of Gödel’s theorem (see detailed text).☜EF597B
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Entered by:-
David Price
NodeID:
#1189
Node type:
SupportiveArgument
Entry date (GMT):
9/5/2006 11:14:00 AM
Last edit date (GMT):
12/8/2007 6:44:00 PM
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