User:Boris Tsirelson/Sandbox1: Difference between revisions

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===Defined or undefined===
===From technical to human: definitions===


Facts are formulated via notions.
The gap between a bare hardware and a nice application is too wide for a single jump, or even a triple jump (hardware – operating system – programming language – application). Bridging the gap is a laborious task for many programmers. They compose programs of modules, and modules from subroutines. Each subroutine reduces a bit more useful task to a bit simpler tasks. Ultimately, a useful (or even fascinating) task is reduced to the technical instructions of the bare hardware.


In the non-axiomatic approach, notions are nodes of a network whose connections are definitions. If some notions are forgotten they probably can be restored from the others.
Likewise, mathematicians bridge the wide gap between useful notions (say, "ellipse" or "normal distribution") and the primitive notions by a large and complicated system of definitions. Each definition reduces a bit more useful notion to a bit simpler notions.
 
Searching Google for "define:line" we get "a length without breadth or thickness". Similarly we find definitions for breadth, thickness and so on, recursively. Doing so we would get a large subnetwork; here is its small fragment:
 
*Line: a length without breadth or thickness
**Length: linear extent in space
***Linear: along a line↑
***Extent: a range of locations
****Location: point or extent↑ in space
***Space: unlimited expanse in which everything is located↑
**Breadth: the extent↑ from side to side
***Side: a surface forming part of the outside of an object
****Surface: the extended↑ two-dimensional outer boundary of a three-dimensional object
**Thickness: the dimension through an object as opposed to its length↑ or width
***Width: the extent↑ from side↑ to side↑
 
(Up arrows mean: see above.) We observe that
* circularity appears routinely; for example: line→length→linear→line;
* the definition of a single notion involves recursively a large number of other, quite remote notions.
 
Such system of notions is unsuitable for a mathematical theory. Here, circularity is disallowed, and the set of involved notions is kept reasonably small (whenever possible).
 
In the non-axiomatic approach, notions are a tower of defined notions, grounded on the basis of more fundamental notions called undefined primitives. If all defined notions are forgotten they surely can be restored from the undefined primitives. The undefined primitives are sparse and simple, not to be forgotten.

Revision as of 07:04, 18 June 2010

From technical to human: definitions

The gap between a bare hardware and a nice application is too wide for a single jump, or even a triple jump (hardware – operating system – programming language – application). Bridging the gap is a laborious task for many programmers. They compose programs of modules, and modules from subroutines. Each subroutine reduces a bit more useful task to a bit simpler tasks. Ultimately, a useful (or even fascinating) task is reduced to the technical instructions of the bare hardware.

Likewise, mathematicians bridge the wide gap between useful notions (say, "ellipse" or "normal distribution") and the primitive notions by a large and complicated system of definitions. Each definition reduces a bit more useful notion to a bit simpler notions.

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