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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 M - Map

 

Map: Representation of some features and relations in some territory; in mathematics: function with specified domain and range. A.k.a. mapping.

The ideas of a map and the closely related mapping are very fundamental, and are somehow involved in much or all of human cognition and understanding - which after all is based on the making of mental maps or models of things.

The first definition that is given is from the use of "map" in cartography and the second from mathematics, but both are related, and mappings can be seen as mathematical abstractions from maps.

1. maps: It is important to understand that one of the important points of maps (that also applies to mappings) is that they leave out - abstract from, do not depict - many things that are in the territory (or set) it represents. More generally, the following points about maps are important:

       the map is usually not the territory (even if it is part of it) 

       the map does usually not represent all of the territory but only certain kinds of things occurring in the territory, in certain kinds of relations

       the map usually contains legenda and other instructions to interpret it

       the map usually contains a lot of what is effectively interpunction

       maps are on carriers (paper, screen, rock, sand)

      the map embodies one of several different possible ways of representing the things it does

      the map usually is partial, incomplete and dated - and

      having a map is usually better than having no map at all to understand the territory the map is about (supposing the map represents some truth)

      maps may represent non-existing territories and include guesses and declarations to the effect "this is uncharted territory"

It may be well to add some brief comments and explanations to these points

Maps and territories: In the case of paper maps, the general point of having a map is that it charts aspects of some territory (which can be seen as a set of things with properties in relations, but that is not relevant in the present context).

Thus, generally a map only represents certain aspects of the territory it charts, and usually contains helpful material on the map to assist a user to relate it properly to what it charts.

And maps may be partially mistaken or may be outdated and still be helpful to find one's way around the territory it charts, while it also is often helpful if the map explicitly shows what is guessed or unknown in it.

2. mappings: In mathematics, the usage of the terms "map" and "function" is not precisely regulated, but one useful way to relate them and keep them apart is to stipulate that a function is a set of pairs of which each first member is paired to just one second member, and a map is a function of which also the sets from which the first and second members are selected are specified. (These sets are known respectively as domain and range, or source and target. See: Function.)

Note that for both functions and maps the rule or rules by which the first members in the pairs in the functions and maps need not be known or, if it is known, need not be explicitly given. Of course, if such a rule is known it may be very useful and all that may need to be listed to describe the function or map.

Here are some useful notations and definitions, that presume to some extent standard set theory. It is assumed that the relations, functions and maps spoken of are binary or two-termed (which is no principal restriction, since a relation involving n terms can be seen as pair of n-1 terms and the n-term). In what follows "e" = "is a member of":

A relation R is a set of pairs.
A function f is a relation such that
   (x)(y)(z)((x,y) e f & (x,z) e f --> y=z).
A map m is a function f such that
   (EA)(EB)(x)(y)((x,y) e f --> xeA & yeB).
That m is a map from A to B is also written as:
   "m : A |-> B" which is in words: "m maps A to B".

There are several ways in which such mappings can hold, and I state some with the usual wordings:

m is a partial map of A to B:
    m : A |-> B and not all xeA are mapped to some yeB.
m is a full map of A to B:
    m is a map of A to B and not partial.
m is a map of A into B:
   m : A |-> B and not all yeB are mapped to some xeA.
m is a map of A onto B:
   m : A |-> B and not into. 

One reason to have partial maps (and functions: the same terminology given for maps holds for functions) is that there may well be exceptional cases for some items in A. Thus, if m maps numbers to numbers using 1/n the case n=0 must be excluded.
 


See also: Function, Mathematics, Representing


Literature:

Griffith & Hilton

 Original: Jan 28, 2006                                                Last edited: 12 December 2011.   Top