February 8, 2013

Philosophy: On representing, maps and meanings
On representing
On maps
3. On meaning
4. On formalizations of representing
About ME/CFS


Yesterday I wrote about the
crisis (<- link to 152 Nederlogs on the subject since September 1, 2008); today about one of the most fundamental human capacities, namely the capacity of - symbolical - representing arbitrary real, imaginary and impossible things, that is in the present Nederlog represented by four entries in my Philosphical Dictionary.

I believe that the capacity of representing and its adequate understanding are fundamental for understanding what it is to be human, and indeed for understanding what human understanding is (representation), and I believe that I present something that is in part original, while I know the formalities will not make it more popular, but these does get explicated in the remarks that follow, and in
my Philosophical Dictionary.

The reasons to quote this today are that my eyes are not too well, nor am I, and I wanted to change the subject, while this is at hand and important.

Three good books that deal with aspects of representing, of which at least two are not as well known as they ought to be are Ogden & Richard's "The meaning of meaning"; Engell's "The creative imagination" and Tough's "The development of meaning". [1]

Most of what follows should be mostly self-explanatory if you follow the links, that are all to the
Philosphical Dictionary. And there is an endnote.

1. On representing

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.

See: Formalizations of representing, Map .

On maps

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.

3. On meaning

Meaning: What a term, statement, symbol, gesture or sign refers to or represents.

This is a very important notion, and not easy to explain well. Two good explanations in book form are Ogden & Richard's "The meaning of meaning" and Lyons' "Semantics".

One important ambiguity about the term "meaning" should be noticed to start with, since it vitiated quite a lot of analytical philosophy. It is this: By the meaning of a term of statement, such as "elephant", one may refer to either some of the ideas or concepts people may have about elephants, such as mental pictures or criterions by which to recognize elephants, or some real elephant(s). The same goes for statements, like "I saw an elephant in the zoo".

A way of keeping these apart is to write "elephant" for the term, 'elephant' for the idea or concept, and elephant - without quotation marks of any kind - for the thing one calls by the term "elephant" and may remember by the idea 'elephant'.

Unless I say so explicitly, in general I will use the term "meaning" to refer to ideas or concepts one may have, rather than the entities one's ideas or concepts represent in some reality, if only because often one does not know whether there really is something as one means by the term, and because one must be able to understand the meaning of a term or statement before one can sensibly make up one's mind whether one believes it to represent anything real (in the sense of "real" one uses).

4. On formalization of representing.

Formalizations of representing: There are many formalizations of the fundamental idea of representing, that may be defined as

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.

Now let's try to make this more precise and general with the help of set theory:

Suppose $ is a society, I is a set of ideas, D is a domain, and L is a language. If S is a set, S* indicates its powerset. A language is here identified with its set of terms, and it is presumed the language contains predicates and subjects.

Also, a domain or universe of discourse consists of anything one may have ideas about, whether real or unreal, true or false, or containing much or little.

Then I define four variants of "In society $ function i helps to
represent domain D by ideas I" using a language L, where the function is the above correlation. I first list them with minimal explanations, and then give some comments.

Representing domains by ideas:

r($, i, D, I) IFF i : D* |-> I* &
                     (ae$)(I inc a) &
                     (deD)(Dj inc D) (deDj iff i(d) e i(Dj))

In words:

In society $ function i helps to represent domain D by ideas I iff i maps the powerset of D onto the powerset of I and I is included in every member a of $ and for everything d in D and every subset Dj of D, d is an element of Dj iff the i of d is an element of the i of Dj.

Clearly, the fundamental point of the definition is the equivalence in the last conjunct, that relates statements about the domain with statements about ideas about the domain.

Note that here and in the later definitions set theory is used to define the notion of representing in various forms, so that in effect the notion of representing is represented set-theoretically, and that this involves an assumption to the effect that the domain D and the ideas I are fairly considered as sets or classes of things.

Also note that the notion of powerset is used to make sure that all the possible distinctions that can be made set-theoretically can be rendered in the presumed equivalence that is the kernel of representing. [2]

Representing ideas by language:

r($, j, I, L) IFF j : I* |-> L &
                    (ae$)(L inc a & I inc a) &
                    (xeI)(Ik inc I) (xeIk iff (EPeL)(EseL)(j(x)=s & j(Ik)=P & Ps) )

The translation is similar to the one given above, and so is the main point of the definition.

The difference with the previous definition is that here ideas are correlated with the terms of a language, that is supposed to have predicates and subjects.

Representing language by ideas:

r($, m, L, I) IFF m : L |-> I* &
                      (ae$)(L inc a & I inc a) &
                      (PeL)(seL)(Ps iff (ExeI)(EIk inc I)(m(s)=x & m(P)=Ik & xeIk ) )

This is the converse of the previous definition, and may be taken to involve or explicate the notions of meaning and linguistic truth: The statement that something called s has a property called P is - linguistically - true, in effect, if whatever is meant by s belongs to the set of whatever is meant by P.  

The reason to insert "linguistically " is that even if it is, say, a linguistical and ideal truth that whales are fishes, this may be false in the domain of facts. To establish that one needs the converse of the first definition:

Representing ideas by domains:

r($, d, I, D) IFF d : I* |-> D* &
                      (ae$)(I inc a) &
                      (xeI)(Ik inc I) ( xeIk iff d(x) e d(Ik))

As I remarked, this is the converse of the first definition and may be taken to involve or explicate the notions of denotation and factual truth: The idea that something x is an Ik is true if and only if whatever x stands for belongs to the set of whatever Ik stands for.

Before extending these definitions by including the notion of probability, it may be well to make a few remarks on the terms I introduced.

  • Every human being (rare exceptions excluded, who tend not to acquire a language at all) is educated in some human society in which he or she learns some language, and
  • every human being may be credited with having quite a few ideas that are much like those of other human beings.

This accounts for the references to a society $ and a language L, about which I will say a little more below.

  • In each definition, the language L and the ideas I are, where appropriate, asserted to be a subset of every member of the society $ - which means that all of these definitions are somewhat idealizing.
  • The four functions introduced - i, j, m, and d - refer to correlations one learns when learning a language L and what its terms are used to refer to. It is much harder to explicate precisely what this must be like than to assume it exists and has the property claimed by the equivalences.

Finally, one part of the reason to explicitly refer to a society is that learning a human language happens in a human society, and another part of the reason to do so is that then one can make a number of distinctions, assumptions and definitions that cannot be made without it.

All of the above can be taken probabilistically which then generalizes the above. To do so it is convenient to introduce the notion of approximate equality as in "(p(xeXi) ≈ p(f(x)ef(Xi))" where "≈" is taken as "differs no more than e from" i.e. "0 <= | p(xeXi) - p(f(x)ef(Xi)) | <= e", and where e is some convenient small number.

Here e is clearly itself between 0 and 1, and if it is 1 the asserted approximate identity is useless since it conveys no information (as the difference between two probabilities is never larger than 1). However, an advantage of introducing probabilities is that probabilified propositions admit of more subtle analyses and distinctions.

The above four definitions using probability on the plan just sketched are as follows: 

r($, i, D, I,) IFF i : D* |-> I* &
                      (ae$)(I inc a) &
                      (deD)(Dj inc D) (p(deDj) ≈ p(i(d) e i(Dj)) )

r($, j, I, L) IFF j : I* |-> L &
                    (ae$)(L inc a & I inc a) &
                    (xeI)(Ik inc I)(EPeL)(EseL) (p(xeIk) ≈ p(Ps) & j(x)=s & j(Ik)=P)

r($, m, L, I) IFF m : L |-> I* &
                     (ae$)(L inc a & I inc a) &
                     (PeL)(seL)(ExeI)(EIk inc I) (p(Ps) ≈ p(xeIk) & m(s)=x & m(P)=Ik))   

r($, d, I, D) IFF d : I* |-> D* &
                      (ae$)(I inc a) &
                      (xeI)(Ik inc I) ( p(xeIk) ≈ p(d(x) e d(Ik)) )

Most of what needs to be said about these definitions has been said when presenting their non-probabilistic form, and one can see that even if e is rather large, say 1/2 or 1/4, one may have ideas about the real probabilities of facts and things that are adequate enough to help one guide one's decisions.

We can now also use LPA and express and consider:

(*) (ae$) aB (be$) bB (Ei)(Ej)(Em)(Ed)(EI)(ED)(EL)
                              (r($, i, D, I) & r($, j, I, L) & r($, m, L, I) & r($, d, I, D))

that is:

All members of society believe that all members of society somehow share ideas about the representation of domains, ideas and language, and about the meanings and denotations of terms.

Here the "somehow" refers to the fact that most members of society would find it hard to specify by what functions they relate ideas, domains and expressions, even if they know quite well how to do it - and indeed (*) claims no more than that the required functions exist.

Also, it may be observed that in fact these functions are stipulative and symbolic and what matters are the defining properties that insist on certain kinds of equivalences (or approximate identities when probabilities are used).

Furthermore, it should be noted that (*) expresses - even while it attributes to all members of society $ a belief about all members of society $ - a minimalistic idea, in that (*) does not imply any agreement on the ideas, language, domain and functions used to correlate their items between a and b or any other members of $.

What in fact seems to be attributed by members of a society, which may be taken as small as a family in which a toddler is learning a language and trying to reach some general assumptions about doing so, is the following rather stronger assumption:

(**) (ae$) aB (Ei)(Ej)(Em)(Ed)(EI)(ED)(EL) (be$) bB
                              (r($, i, D, I) & r($, j, I, L) & r($, m, L, I) & r($, d, I, D))

that is - and here it may help to think of the society as a family:

All members of society believe that there is a language, a domain, and a set of ideas with appropriate functions such that all members of society somehow share ideas about the representation of the domain, the ideas and the language, and about the meanings and denotations of terms.

For at least to those who learn the language, it will thus be represented, and indeed in any society there are many ideas and experiences that are - it would naturally seem - shared by all members of the society.

Also, if one takes the society $ small enough - say, one's family and friends - both (*) and (**) will be true, and indeed there will be quite a lot of shared beliefs that count as presumptive knowledge.

If one takes the society larger, what one shares with all others in it in terms of beliefs will be less, but even so all speakers of the same language share many ideas about that language and what its terms mean, while all humans that know some language may be taken also to share quite a few ideas, namely at least about language in general, and about human beings in general, and the things all humans must do and know in order to survive and function as a member of some society.

See also: Fantasy, Imagination, Rationality, Realism, Reason, Representing, Science

Endnote: These and related subjects - representing, meaning, logic and such - I would like to write (much) more about, but often don't because I need more energy to do it well, while there also seem to be very few who are really interested in these subjects, especially if they are (also) treated with formal methods.

And while I agree it can be done without set theory or logic, I think one or the other is needed to do it well, and not to get caught by subtleties.

I spent a lot of time reading about representation and meaning, but found it is mostly not treated well, according to my understanding - and perhaps I will be so bold as to try to write out something like "Basic formal logic and formal philosophy" in Nederlog.

If I do this at all, this is the only way to do it - piecemeal, if and when I can and feel like it - but I make no promises, and I also have no illusions this will make Nederlog more popular.

It is something I would like to do if I can, if only to have written out these fundamental things as clearly as I can. [3]

[1] The first two are still available on the internet in some form.

In fact I do not assume infinities: All that seems required for representing, at least for most ordinary cases, and next to natural language, are finite sets, and the basic notions of (proper) names (of indivuals), nouns or common names (of collections of individuals), predicates (the part of a statement that are not names or nouns), and the  logical terms and, not, is the same as, and such that. These allow the definitions of the basics concepts and terms of logic and set theory.

[3] Which really is considerably less well, and much less easily, then would be the case if I were healthy, which also tends to obstruct me - but then that is how things are for me. So I'll see what I can do, along the lines of
"Basic formal logic and formal philosophy" in Nederlog.

About ME/CFS (that I prefer to call M.E.: The "/CFS" is added to facilitate search machines) which is a disease I have since 1.1.1979:
1. Anthony Komaroff

Ten discoveries about the biology of CFS(pdf)

3. Hillary Johnson

The Why  (currently not available)

4. Consensus (many M.D.s) Canadian Consensus Government Report on ME (pdf - version 2003)
5. Consensus (many M.D.s) Canadian Consensus Government Report on ME (pdf - version 2011)
6. Eleanor Stein

Clinical Guidelines for Psychiatrists (pdf)

7. William Clifford The Ethics of Belief
8. Malcolm Hooper Magical Medicine (pdf)
Maarten Maartensz
Resources about ME/CFS
(more resources, by many)

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