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For a [[theory (mathematics)|mathematical theory]], correctness means formalizability.
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''"In practice, the mathematician ... is content to bring the exposition to a point where his experience and mathematical flair tell him that translation into formal language would be no more than an exercise of patience (though doubtless a very tedious one)."''<ref>{{harvnb|Bourbaki|1968|loc=page 8}}.</ref> Reliability of these experience and flair appear to be high but not perfect. Formalization is especially desirable in complicated cases, but feasible only in very simple cases, unless computers help. Similarly, without computers a programmer is able to debug only very simple programs.
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.


A '''proof assistant''' is a computer program used interactively for developing human-readable reliable mathematical documents in a formal language.
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).
 
Nowadays (about 2010) the most successful project of this class combines
* ''Isabelle,'' a generic system for implementing logical formalisms;
* ''Isar'' (Intelligible SemiAutomated Reasoning), a versatile language environment for structured formal proofs;
* ''Proof General,'' a configurable user interface (front-end) for proof assistants;
* ''HOL'' (Higher-Order Logic).
 
This combination is meant below (unless otherwise stated explicitly).
 
==Example session==
 
An existing file, verified before, is used as the source text in this example. Thus, the session is not really interactive. However, this fact is not known to Isabelle. The proof assistant treats the session as interactive, and the text as created anew.
 
[http://www.cse.unsw.edu.au/~kleing/top100/#1 Top 100 theorems in Isabelle] +
[http://www.cs.ru.nl/~freek/100/ Formalizing 100 Theorems] +
[http://isabelle.in.tum.de/ Isabelle] +
[http://isabelle.in.tum.de/download.html Download and installation] +
[http://isabelle.in.tum.de/projects.html Projects] +
[http://isabelle.in.tum.de/library/ The Isabelle2009-2 Library] +
[http://isarmathlib.org/ IsarMathLib: A library of formalized mathematics for Isabelle/ZF]
 
{{Image|Isabelle1.png|right|350px|The graphical user interface started.}}
 
 
 
 
 
 
The graphical user interface started.
 
 
 
 
{{Image|Isabelle2.png|left|350px|The source file is read.}}
 
 
 
 
&nbsp;&nbsp;The source file is read.
 
{{Image|Isabelle3.png|right|350px|Definitions are processed; the formulation of the first lemma is being processed.}}
 
 
 
Definitions are processed; the formulation of the first lemma is being processed.
 
{{Image|Isabelle4.png|left|350px|The formulation of the first lemma is processed; the goal is pending.}}
 
The formulation of the first lemma is processed; the goal is pending.
 
{{Image|Isabelle5.png|right|350px|At last, a really serious lemma.}}
 
{{Image|Isabelle6.png|left|350px|The goal.}}
 
{{Image|Isabelle7.png|right|350px|"We have", why?}}
 
{{Image|Isabelle8.png|left|350px|Here is why!}}
 
{{Image|Isabelle9.png|right|350px|Really complicated arguments...}}
 
{{Image|Isabelle10.png|left|350px|Happy end is coming.}}
 
{{Image|Isabelle11.png|right|350px|The theorem is proved.}}
 
{{Image|Isabelle12.png|left|350px|Some options of the Proof General.}}
 
{{Image|Isabelle13.png|right|350px|Isabelle, show us your methods...}}
 
{{Image|Isabelle14.png|left|350px|...and your term bindings...}}
 
{{Image|Isabelle15.png|right|350px|...and the theorems.}}
 
==Notes==
{{reflist|2}}
 
==References==
 
{{Citation
| last = Bourbaki
| first = Nicolas
| title = Elements of mathematics: Theory of sets
| year = 1968
| publisher = Hermann (original), Addison-Wesley (translation)
}}.

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


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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