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David Hilbert aimed to find axioms sufficient for all mathematics and to prove their consistency from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent. A fatal blow was dealt by the second Gödel's [[incompleteness theorem]]. Consistency of a theory cannot be proved by a weaker theory, nor by the same theory. It can be proved by a stronger theory, which does not dispel doubts: if the given theory is inconsistent then the stronger theory, being all the more inconsistent, can prove every claim, be it true or false.
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


Many mathematicians feel that specialized theories, being more reliable than universal theories, are like watertight compartments. If a contradiction will be found in the used universal theory, specialized theories will separate and wait for a better universal theory.
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
 
<blockquote>I have always felt that, if one day someone came up with a contradiction in mathematics, I would just say, "Well, those crazy logicians are at it again," and go about my business as I was going the day before.<ref>Vaughan Jones. See {{harvnb|Casacuberta|Castellet|1992|loc=page 91}}.</ref></blockquote>
 
<references/>
 
==References==
 
{{Citation
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| first = Émile
| title = Probabilities and life
| year = 1962
| publisher = Dover publ. (translation)
}}.
 
{{Citation
| last = Bourbaki
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| title = Elements of mathematics: Theory of sets
| year = 1968
| publisher = Hermann (original), Addison-Wesley (translation)
}}.
 
{{Citation
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| title = The character of physical law
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| year = 1995
| publisher = the MIT press
| isbn = 0 262 56003 8
}}.
 
{{Citation
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{{Citation
| last = Mathias
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| year = 2002
| title = A term of length 4,523,659,424,929
| journal = Synthese
| publisher = ?
| volume = 133
| issue = 1/2
| pages = 75–86
| url = http://www.springerlink.com/content/x28504221108023t/
}}.
(Also [http://personnel.univ-reunion.fr/ardm/inefff.pdf here].)
 
{{Citation
| year = 1992
| editor-last = Casacuberta
| editor-first = C
| editor2-last = Castellet
| editor2-first = M
| title = Mathematical research today and tomorrow: Viewpoints of seven Fields medalists
|series = Lecture Notes in Mathematics
| volume = 1525
| publisher = Springer-Verlag
| isbn = 3-540-56011-4
}}.

Latest revision as of 03:25, 22 November 2023


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The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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