I continue being not well, and otherwise also as before,
so I cannot do much.
However, now and then I can do a little
1. More comments on Wittgenstein
2. Some site news
comments on Wittgenstein
Being a philosopher, I felt inspired to take up
my edition and notes on
Tractatus (Logico-Philosophicus). '
The links just given are to the
Wikipedia; and this is
beginning of my translation and comments, with here the complete text of the
and here my
Today there is more of the same - and I
should start by remarking as of today it is all done in a new directory
for the Tractatus, which means that all the links to the Tractatus with
my comments differ from what they were until today.
I have now uploaded the beginning of the
new version for the theses starting with 3 and also some more summaries
to the theses starting with 2. For
those who are interested, here is my summary for
2.1.., as it
happens on a subject that
presents his picture theory of meaning, as it became known.
It is based on
a sound intuition, namely that human beings try to understand how things
are by trying to imagine what they, their causes or antecedents, and
their effects or consequences really are.
presentation of it is unclear, and also mistaken: He seems to confuse,
at least for the most part, language and ideas (images, fantasies), and
also true ideas (images, fantasies) and the real situations these
represent, while his picture theory of meaning, rather oddly, seems to
presuppose that language is hieroglyphic or ideographic rather than
alphabetic and symbolic.
hieroglyphic or ideographic language, the written form does not
represent sounds of speech by arbitrary letters and words composed of
these, which in turn, to be understood, must evoke an idea of what is
meant, but instead consists of stylized pictures of things and
situations that represent.
or ideographic language thus is closer to how the human mind represents,
in the sense that (some, not al) ideas are in some ways rather like
hieroglyphs or ideograms for whatever they stand for, than is an
alphabetic language, but an alphabetic language, with letters and words
representing sounds of speech rather than things meant by speech, is
much easier to learn, to write and to use.
theory of meaning is one of the main ideas of the Tractatus, and it is
clearly mistaken - and incidentally also shows how confused e.g. Russell
and Wittgenstein must have been about the relations of language and
symbolism to reality, having both believed at least for some years this
fundamentally confused and hieroglyphic theory of symbolic
representation makes sense, and is an adequate foundation on the basis
of which one can explain language, logic, and mathematics.
Here are some
points from my notes, with the sources linked in.
On the term "picture":
term I have translated as "image" is "Bild". Other translators of the
Tractatus have translated "Bild" as "picture" (which gave rise to
phrases like "Wittgenstein's picture theory of meaning").
"image" for three reasons over "picture".
is a close parallel between "image" in the sense of "picture" and
"image" in the sense of "set associated by a
This is also the case in German, where one speaks of "das Bild einer
Funktion" where in English one speaks of "the image of a function", and
W. certainly knew this.
seems closer to W.'s intended epistemological use of the term "Bild".
W.'s basic leading idea here comes, directly or indirectly, from Hertz,
and it is that we know the world by making a mental model of it, rather
like we know the lay of the land if we have a mental or paper map of it,
and like we may know what to expect about the behaviour of a real
airplane in turbulence if we have investigated model airplanes in wind
Thus, we know
the world by imagining a model for it, which is correct to the extent
that the model is an image (in the mathematical sense) of the world,
and, at least in part, conversely.
And the reader
should note that Hertz's theory of representation, first stated in 1899
in Heinrich Hertz's posthumous "Principles of Mechanics" in fact
both inspired Wittgenstein's theory and is much clearer and more
sensible, for Hertz clearly stated that we know the world by making a
mental model of it, rather like we know the lay of the land if we have a
mental or paper map of it, which seems quite correct, but not in a
pictorial or hieroglyphic way, except in some simple cases.
On representing and maps
important to realize that being an image tends to be a matter of degree
maps are more or
less detailed and have more or less accuracy, but that on the other hand
to be able to be an image,
model or map of
something at all there must be some correspondence between the elements,
properties and relations of the things making up the image, model or
map, and the things, properties and relations it
Here are some
useful points about maps:
some features and
relations in some
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.
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:
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
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.
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:
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
relation R is a set
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.
On human experience
In a sound and
intuitive sense our
falls apart into the three fields of
We sense our environments and our bodies; we remember what happened to
us and in the world we sensed; and we imagine what has happened, happens
or might happen, essentially by altering some of our memories in certain
ways and for certain reasons, and proposing the altered memories as
models for certain
In each of
these cases we design a mental model of something
real or unreal;
in each of these case we assume again a correspondence between the model
and whatever it is a model of; and in each of these cases the
correspondence is plausibly explained as a homomorphism between model
and modeled. (This means that the two things have a similar structure in
exemplification of this are a map and its territory (i.e.
whatever the map is a map of) and important things to keep in mind are
that the territory may be far more complicated than its map (in practice
it is, for else there is little point to having a map), and that maps
are adequate by reference both to purposes and to territory:
A map is
adequate for a
given purpose if
it corresponds sufficiently well to its territory to enable its users to
realize their purpose when in the territory.
it should be noted that by far the greatest part of what human beings
think of is imaginary, and not known to be true by them, basically
because it consists of guesses and generalizations rather than precisely
Logical space and domains
It seems that
by "logical space" W. meant something like domain or universe of
discourse, but he is not very clear, and what he does say seems to be
generalized from truth tables, as in "the logical space involved in "it
rains (T) or it is cold (C)" consists of [(R&C) V (R&~C) V (~R&C) V
(~R&~C)], that is, a tautology involving all mentioned possibilities in
all possible combinations.
What is true
apart from what W. meant is that the use of language generally does
involve some universe of discourse, that is some supposed reality in
which the terms of the language represent or do not represent some
thing(s) and the statements of the language represent or do not
represent some fact (or event or structure or process) in that supposed
Part of what
is involved for logic and mathematics, and to some extent natural
language, was later articulated in terms of model theory, which I merely
mention but do not treat here, and it should also be mentioned that the
same text, statements or terms may be taken to different universes of
discourse, that also need not be the real world, but may be an imagined
reality known to be not really so (as in "Sherlock Holmes lives in Baker
Street, London") or an imagined reality that is, at least as far as
one's knowledge goes, hypothetical.
Images, models and ideas
It is true -
in a very Pickwickian sense also - that images often represent or are
taken to represent some reality, that may be imaginary or hypothetical,
but most human ideas do not represent like images do, which one may take
to happen in a pictorial way, but in a more complicated way.
restricting oneself to images, it should be noted that images may depict
impossible situations: Something possible may be an image of
something impossible, apparently because to be an image requires
being a physical object
together with certain principles of
that jointly produce the image, and impossible images seem to depend on
a confusion or identification of some of the properties of the physical
object and some of the properties of the thing imagined using the
physical objects and the principles of representation.
Thus a number
of impossible 3-dimensional objects, like Escher's "Waterfall", have
been drawn on basically 2- dimensional paper, using the properties of
planes and the rules of perspective.
Representation, interpretation and reality
phrase for "principles of representation" is "principles of
interpretation", and one important consequence of this is that,
whereas it often seems to us that such-and-such "obviously", "evidently"
or "naturally" means, represents or stands for so-and-so, in any case of
representation the things that do the representing do NOT represent
whatever they represent "obviously", "evidently" or "naturally", but by
some set of
that need to be known and cannot be
the representing things. If this is "evident" etc. this is so only
because this kind of image is habitually interpreted in this "evident"
Nothing represents by itself; whatever represents represents because of
certain assumptions and conventions that connect it to whatever it is
taken to represent.
This is also
the case for perception: Each young animal has to learn to make sense of
its sensations and, for example, learn to read, so to speak, distances
and speeds from seen sizes and movements of shapes, and to relate the
information from one sense, e.g. one's body image, to another, such as
Representing involves principles of representation
I remarked that nothing represents by itself; whatever represents
represents because of certain assumptions and conventions that connect
it to whatever it is taken to represent. Accordingly, [2.141] at least
makes a mistaken suggestion: That one can know that a certain fact X is
an image of another fact Y simply by being acquainted with the fact X.
This is not so in general, for one normally also at least needs to know
that are used to understand what X represents.
It may be so
in a handful of particular cases, like mirror-images, realistic
pictures, good imitations and so on, and it may be that historically
these where the first
symbols, but even
there assumptions are required, if only to the effect that when two
things appear similar in some respect one of these may be used
voluntarily and purposively to evoke an idea of the other.
Signs and symbols
something that is naturally associated with a thing, like smoke is a
sign of fire, and clouds a sign of rain. Accordingly, if one experiences
the sign one expects what it is often associated with.
something that is
associated with a thing, such that the symbol becomes the sign for the
idea of the thing. This seems to require a step beyond signs that no
animal except man has made: To understand that one can agree to use any
(easily reproducible) act or fact as a conventional sign for anything
one wishes to make someone think of.
Representing is more than a simple morphism
There is nothing like an isomorphism or homomorphism of a picture (usually
2-dimensional) or text and what it represents, and the manner of representation
depends always on some human assumptions and usually on many, while these
assumptions are rarely adequately summarized. This is also the reason W. missed
"The structure of the picture", whatever it may be - a diagram, a picture by
Escher, a stylized map of the London Underground, a 2-dimensional photograph of
a 3-dimensional object, a trick "photograph" of an impossible object, or
whatever - is generally NOT "the
structure of what is
pictured", but at best
a structure that, together with
assumptions that are
not given with that structure, a key to whatever structure the depicted thing(s)
may really have.
(2.151) Even a
realist picture, even a tromp d'oeil, are 2 dimensional things
representing 3-dimensional things, generally (if not necessarily) embodying some
unarticulated rules of
Representing is often of uncertain or unexisting possibilities
One of the things W. mostly misses is that representing is often of uncertain or
unexisting or merely fantastic, imaginary or hypothetical possibilities, at
least as far the person representing knows.
K. (2. 1514)
Representing conventions, legenda etcetera
Rarely the pictorial or
relationship(s) - whatever W. precisely means (and I fear he didn't mean
anything precisely, for then he would have seen it is not so) - is given
with the picture, diagram,
map, statement or formula.
Usually it is unstated and tacit, and in the few cases something like it is
given with the picture, as with legenda on a map in cartography, it is quite
Even pictures often represent in a very distorted way
Pictures - look at children's drawings - often are partial and distorted,
whether by inability or on purpose, as in a charicature of a politician's face,
or as with a picture of an arrow and a pointing finger. Besides, it would seem
some pictures, indeed including charicatures, but not limited to these, may be
ironical, and depict by distortion or irony - as did Hogarth, for example.
(2.161) What is at
best true, and that mostly of "typical" pictures, as e.g. occur in children's
books, is that there is a similarity between some aspects of what is depicted
and some aspects of what depicts it.
W.'s "pictorial form" is mostly word magic
W.'s phrase "pictorial form" is a manner of verbal suggestion of what is not so
or at least not sufficiently clarified to be presented with rational conviction.
Even pictures do generally not represent photographically
In almost all ordinary pictures - children's drawings again come to
mind - there is no simple isomorphism between picture and depicted,
and the relation between picture and things pictured is far more
complicated than a simple likeness of the form "circle here so circle
Wittgenstein on pictures and pictorial form is rather contradictory
W. on pictorial form is contradictory and misleading: First, he
insists pictures and pictured must be alike, and then he insists in
2.172 that this presumed likeness of pictures and pictured, which
exists in simple prototypical cases, as pictures of triangles tend to
be triangular and pictures of circles circular "cannot,
however, depict its pictorial form".
In fact, this is related to W.'s later doctrine of what cannot be
said but only be shown, and this first occurence of it is just
inconsistent with what he claimed before about picturorial
"Logical form" as word magic
W. 's suggestion that a picture represents becausse it has
something like "logical
form", which it also - in some deeply mysterious sense - is "the
form of reality", is cant : At best, it amounts to the far from
deep theses that pictures and things depicted tend to look alike,
especially in prototypical cases. That this is anything like "logical
form" as in the Principia Mathematica is pretensious word magic.
is a gross form of this: "A
picture whose pictorial form is logical form is called a logical
picture" and the same goes for (2.182)
That "Logical pictures
can depict the world" is mostly cant
For why would the same not be true of illogical or non-logical or
any kind of pictures whatsoever?! What is "logical" about a picture
that does depict some aspect of the world truly, according to its
viewers? Why does it need to be "logical" indeed in any sense, and
more specifically in W.' sense of "logical", which is "tautologous" as
Sofar for that summary of my comments on
Wittgenstein's theses starting with "2.1..".
As before, if you can't make much of it, I'm sorry
- I can only make it clearer by making it longer, and it may be that you
need some knowledge of analytic philosophy and of mathematical logic to
follow all or be interested at all.
Have fun with it, or do something else that you like
2. Some site news
Meanwhile I have corrected
some typos in the last Nederlog, and should also add that for the moment my
edition of and comments on the Tractatus is in a state of flux - which may
explain why some links may not work or some backgrounds may be absent. Also, as
I remarked before, the Tractatus + comments are now in a new set of directors
And, at VERY looooong last
- namely after 14 years - I have updated the 5 main indexes of my site
to the effect that
it's meta name="Maartensz" and it's content="philosophy,
ME in Amsterdam,
This is supposed to spread my renown far and wide over the internet, since
searchbots and searchmachines thrive on this.
Well... I have added links to the
subjects in the list. Have fun!