User:Boris Tsirelson/Sandbox1: Difference between revisions

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==Univalent or multivalent==
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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.


Plane geometry (called also "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe. The question "which plane?" is inappropriate, since planes do not differ in their geometric properties. Every two planes α, β are isomorphic, that is, there exists an isomorphism ''f'' between α and β. Treating α and β as sets of points one defines isomorphism as an invertible (one-to-one and onto) map f : α to β preserving all primitive relations. Namely: f maps lines into lines; the distance between f(A) and f(B) on β is equal to the distance between A and B on α; etc. The same is required of the inverse map <math>f^{-1}:\be\to\al</math>.
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).
 
Axioms of the plane Euclidean geometry leave no freedom, they determine uniquely all geometric properties of the plane. More exactly: all Euclidean planes are mutually isomorphic. In this sense we have "the" Euclidean plane. In terms of Bourbaki, the plane
Euclidean geometry is an univalent theory. In contrast, axioms of a linear space (called also vector space) leave a freedom: a linear space may be one-dimensional, two-dimensional, three-dimensional, four-dimensional and so on (infinite dimension is also possible). The corresponding theory is multivalent.
 
According to Bourbaki, the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.
 
A similar idea occurs in mathematical logic: a theory is called categorical if all its models are mutually isomorphic. However, for Bourbaki a theory is embedded into the set theory, while in logic a theory is standalone (embedded into the first-order logic).
 
    Motivated or indiscriminate
 
Monkeys could type into a computer a sequence of hardware
instructions; the computer could execute them; but the result of such
"programming" has almost no chance to be fascinating or
useful. Fascinating computer games reflect human's
predilections. Useful programs reflect human's needs. A computer is
dull for humans unless its software reflects human's life in one way
or another.
 
Likewise, a theorem is of no interest for humans unless it is
motivated in one way or another by human's life. The motivation may be
quite indirect; many theorems "only" help to prove other theorems,
many are appreciated "only" for their aesthetic value, etc. But some
kind of motivation is vital. Indiscriminate stream of logical
consequences of the axioms is not publishable in the mathematical
literature.

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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