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  Feb 9, 2012                  

Logic+Philosophy: About structures, maps and representations

I completely changed subject, and assembled some definitions from my Philosophical Dictionary that may help some to think or formulate a bit more clearly.

The terms I define are of fundamental importance to human understanding and explaining, for that is done in terms of models, that are representations.

Here are some of the relevant terms defined with links:


As indicated, the given definitions are all from my Philosophical Dictionary - that needs some updating and uploading - and are important for philosophy of science, epistemology and methodology. The links in what follows also are to terms in that dictionary.

Structure: Entity made up of diverse parts that remains the same entity if some of its parts are replaced by other parts of the same kind.

This is a fundamental stipulative definition of a fundamental idea that is fleshed out in various ways in various disciplines, like logic, mathematics and linguistics. Structures in some sense are often supposed to be part of reality, and sometimes reality or some part of it is considered to be made up of nothing but structures.

Representing: Something A represents something B if and only if the properties, relations and elements of A are systematically correlated with the properties, relations and elements of B in such a way that - some of - the latter can be inferred from the former for those who know the correlation.

This seems to be a uniquely human abillity in so far as it depends on the human ability to reason with symbols. In logic and mathematics relations that represent are isomorphisms or morphisms.

The idea that something A represents something B is very fundamental and occurs in many forms.

An often useful instance of representing is a map that represents some territory.

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).  [terminological note 1]
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.

Isomorphism: From the Greek: "Having the same form". As spelled out with some modern mathematics: A 1-1 mapping between two sets such that some (structural) relations - the forms - are preserved.

The mathematical statement just given has the following import: Let A1..An and B1...Bn be sets,; let f be a 1-1 function from Ai to Bi for 1<=i<=n; let S be a relation of the elements of A1..An and R a relation between the elements of B1..Bm with m<=n. Then using Cartesian Products from set-theory and first-order predicate logic:

       f is an isomorphism from A1*..*An to B1*..*Bm for S and R
  IFF S(a1 .. an) iff R(f(a1) .. f(am))

In words: IFF a1 .. an stand in relation S precisely if f(a1) .. f(am), that is the values of a1 .. am for f, stand in relation R.

Often the relations S and R are taken to be the same: Then the same relation - such as a shape, form, curve - obtains in B and in A for the elements f maps from A to B. In these contexts, it is often said more colloquially that f is a 1-1 function that preserves relations.

There are many variants, applications and precifisications in various fields of mathematics and also outside it.

And clearly any isomorphism f for S1 and R1 may also be an isomorphism for relations S2 and R2,  .. , and relations Sk and Rk, while what makes isomorphisms important and a kind of representing is that having the isomorphism one can say what is the case if A1*...*An from what is the case in B1*...*Bm or the other way around. In this sense, a map also is an isomorphism of aspects of the territory, if it is a correct map.

Model: (1) Imitation, likeness, analogy of a thing or kind of things. (2) In logic: Set-theoretical representation of the universe of discourse of a language.

In the first sense, models are made for many kinds of reasons. In general, a model abstracts from certain - possibly many - features of the thing modelled, and imitates only some aspects and features of it. In science, models are often used - e.g. models of airplanes in windtunnels - to find out things about whatever is modelled.

Also in the first sense, if metaphorically, it is often said with considerable sense that when one tries to understand something one constructs a mental model for it, in some ways, that may involve language, imagery or diagrams.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.
    J. von Neumann
   (quoted at Mactutor Mathematics Archive)

In the second sense, a model is a set-theoretical construction that is used as an interpretation for a (formal) language one investigates, and model theory is a collection of sophisticated and highly developed techniques and results in mathematical logic.

[note 1] In "(x)(y)(z)((x,y) e f & (x,z) e f --> y=z)" the letters "x", "y" and "z" are variables; "(x)" is the universal quantifier read "for every x"; "e" abbreviates "is an element of", "f" stands for the relation; "(x,y)" and "(x,z)" are ordered pairs; and "-->" is "implies", so the whole statement amounts to "for every x, y, z: if the pairs (x,y) and (x,z) are in f then y equals z", which again is to say that f has only pairs of which the left members occur only in one pair (and so their right members are unique, which is what makes the relation f a function).

A little later "(EA)" abbreviates "There is a set A". The notation is not standard, but then the benefit is that it can be typed in ASCII, unlike the more usual one.

(This is - incidentally - something I should straighten out programmatically, but is a bore, and anyway will not work for all users. The only thing that does work for the most part and the most users is to use graphics, and to put that in place is again a lot of boring work.)

Corrections, if any are necessary, have to be made later.


As to ME/CFS (that I prefer to call ME):
1.  Anthony Komaroff Ten discoveries about the biology of CFS (pdf)
3.  Hillary Johnson The Why
4.  Consensus of M.D.s Canadian Consensus Government Report on ME (pdf)
5.  Eleanor Stein Clinical Guidelines for Psychiatrists (pdf)
6.  William Clifford The Ethics of Belief
7.  Paul Lutus

Is Psychology a Science?

8.  Malcolm Hooper Magical Medicine (pdf)
 Maarten Maartensz
ME in Amsterdam - surviving in Amsterdam with ME (Dutch)
 Maarten Maartensz Myalgic Encephalomyelitis

Short descriptions of the above:                

1. Ten reasons why ME/CFS is a real disease by a professor of medicine of Harvard.
2. Long essay by a professor emeritus of medical chemistry about maltreatment of ME.
3. Explanation of what's happening around ME by an investigative journalist.
4. Report to Canadian Government on ME, by many medical experts.
5. Advice to psychiatrist by a psychiatrist who understands ME is an organic disease
6. English mathematical genius on one's responsibilities in the matter of one's beliefs:

7. A space- and computer-scientist takes a look at psychology.
8. Malcolm Hooper puts things together status 2010.
9. I tell my story of surviving (so far) in Amsterdam/ with ME.
10. The directory on my site about ME.

See also: ME -Documentation and ME - Resources
The last has many files, all on my site to keep them accessible.

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