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From Heisenberg to Gödel via Chaitin
 INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
, 2004
"... In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later ..."
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Cited by 11 (9 self)
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In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
"... ..."
S.: Passages of proof
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 2004
"... Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs w ..."
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Cited by 1 (1 self)
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Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computerassisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical
legitimise the conquests of intuition... J.
, 2008
"... The object of mathematical rigour is to sanction and ..."
In mathematics you don’t understand things. You just get used to them. J. von Neumann From Heisenberg to Gödel via Chaitin
, 2008
"... In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. A ..."
Abstract
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In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle ” which implies Chaitin’s informationtheoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics. 1