Is it because our mental capacities are limited?
How much information can we compress in a definition?
Is any connection between the amount of information we compress in a
definition and our mental capacity?
And how de we create a good definition?
Is it like an algorithm that allows us to recreate the initial
diversity of information?
And what about the variables we have to discard in order to keep the
definition succinct?
If I use an analogy with the human physical capacities, there is an
amount, an average weight that a person can carry and the same is true
about the average number of different items a person can carry at one
time.
In order to be able to carry more items a person has to organize them,
use a "container" but the weight of the container is still limited by
the person's physical capacities.
Is this something similar to what we do with the information?
We organize the information, extract the variables and the connections
between them and create a definition, an algorithm to compact the
information in order to overcome the limitations of our mental
capacities?
There are people that can process more information, others who can
process more items of information at the same time and some who can do
any, both or none of them.
In a physical capacities competition we try to win by using our
abilities to carry more items per load (weight), while in a mental
competition we try to use more arguments, more compressed information
to win.
Whenever we do not organize the information in the same way don't we
usually end up by being unable to come to an understanding and start
calling each other names?
And isn't this a result of our limited mental capacities that force us
to create concepts, definitions, etc.?
And aren't these concepts, definitions imperfect because our mental
capacities limit the amount of information to be compacted in them?
And aren't our discussions shaped by this limited mental capacity?

Just Playing

It seems to me that you need definitions in order to communicate meaning.
If, to take a silly example, I only use one word 'la', so all my utterances
are 'la la la la...' it will be hard for me to communicate any meaning.
It's because I use different words, each with a different definition, that I
can tell others what I see, what I think.

The weakness you see appears to me to be due to not adequately defining, of
not actually having agreed-upon definitions. I think this is more a matter
of limited patience then limited mental capacity, nailing down the exact
definitions is a tedious process. Ed

....snip...

A definition doesn't compress information. You're confusing the symbols
with the referents.

Just Playing

I am sure I make a lot of mistakes and I am confusing terms, but this
doesn‘t seem to stop me from Just Playing.
I guess that what I am trying to understand is the connection between
our mental capacity to process information and the symbols we use to
do it.
I am wondering how much bytes of information we can process per second
and how it affects our way of creating symbols.

Just Playing

A neuron responds in the 100s of milliseconds. IOW, a train of pulses
can be repeated about 10 times per second. (Eg, the neurons (light
receptors) in the eye have a cycle time of about 1/10th of a second; the
illusion of motion in movies and videos depends on this.) The question
is how to interpret the neural response as a bit rate. AFAIK, the only
variation in neural response is the rapidity of the pulses within a
train, which suggests that a train carries two bits (fast/slow). Thus a
neuron operates at a bit rate of about 20.

However, the connection topology of the neurons guarantees that the
processing rate of the brain as a whole, or of subsets of it, will be
many times higher. Consider, for example, that small clusters of neurons
in the visual cortex respond to specific features of the visual field
(sometimes, a single neuron does so.) EG, most of us can pick out a
sought for face in a crowd in a matter of a few seconds at most. This
corresponds to a bit rate of several megapixels a second, as measured by
the (linear) processors used in computers. The usual phrase used to
describe this aspect of the brain is "massively parallel," an
unfortunate term IMO, since it implies a two dimensional array or tree.
In fact, the connection topology of neurons is multidimensional, since
many neurons are connected to many neurons. (See the concept of
fractional dimensions to understand how a network embedded in 3-space
can have a dimensionality higher than 3.)

As for what all this has to do with the human capacity for symbolism: I
don't think bit-rates have anything to do with it. It's the topology of
the network that has something to do with it. IMO, at the level of brain
function symbols are specific firing patterns of specific networks,
whose activity is stimulated by inputs from elsewhere in the brain (and
ultimately, in some but not all cases, by input from sensors.) If this
guess is valid, we ought to find symbolic behaviour in any system with
a sufficiently complex neural topology. Some people claim to have
observed evidence symbolic processing in apes (as distinct from
conditioned responses to arbitrary shapes and colours); if true, such
observations imply that apes have brains roughly as complex as humans.
OTOH, it seems no artificial system is complex enough to exhibit
symbolic behaviour.

BTW, mere "connectivity", as measured in average number of connections
per neuron, doesn't tell us much about the topology of the network.

No, we need definitions in order to all of us understand what are we
talking about. For example, suppose I say "I took my wambalgour for a
walk" What am I talking about? Now, say I define wambalgour as a small
white cat, then the sentence means "I took my small white cat for a walk".

We can compress as much information as needed in a definition, in
fact, if you take advanced maths, you may see that simple words like
"Compact Space" or "Torus" relate to many other terms which, in fact,
have long definitions, and so on, until you arrive to the "primal"
terms (which are terms that are left mathematically undefined)

A good definition is one that is clear, and is posted in the term of
an equivalence; that is, every wambalgour is a small white cat, and
every small white cat is a wambalgour. If there is no equivalence,
then there is no real definition

No, by giving a definition, you already presume that others understand
(at least to some great extent) what you are meaning (namely, with your
definition). There is never any guarantee that we have understood what
you were trying to communicate.

I think definitions can make life easier for those who have already
agreed about what they want to communicate. For example, we both speak
English, and we may believe that our partner understands what we want to
say. We can therefore decide ("define"), for example, to use the symbal
Q to denote the set of rational numbers (assuming we know what we are
talking about), or for example, I might write "X1,...,Xn: iid" and tell
you that whenever I write this, I want to say: "random variables X1 to
Xn are idependently identically distributed".

But there is no way to communicate the "meaning" of this. What it comes
down to is that all those words you see now, and that I am writing at
this very moment, are not much better than your "la la la la..."
example. I mean, what do you see? Only letters. The "meaning" is already
there, in your mind, and you are projecting it on otherwise useless letters.

Definitions are a kind of abbreviation, nothing else.

Best regards,
Markus.

This leads to nothing, because "my small white cat" might mean exactly
nothing to me, or something completely different that it does to you. I
might be afraid whenever I think of cats (well, I'm not), or think about
red when you say white or anything else, depending on my experience.

Do not believe that there is a "per se" meaning in any word. When I say
something, you associate your memories to the sound of the words, and
what you see are the pictures in your mind. It's totally subjective.

What would such a "primal" term be then? A definition in mathematics is
an abbreviation for something else, but when giving a definition, the
writer MUST assume that the reader understands most of what she (the
writer) is talking about.

Best regards,
Markus.

Basically yes. This can be compared to our sense perceptions.
For instance, we see much more information about our environment
than is brought to our conscious awareness. The environment has
a close to infinite amount of detail, like a fractal, which exceeds the
storage capacity of the brain.

The method of extracting sufficient and meaningful information by
filtering out the non-essential when this content and priority level
changes from moment to moment it what the human brain does
well enough to deliver survival behavior to the human species.

The difficulty of designing a program that accomplishes this
selection is usually known as the Frame Problem or a variation
of that name, how to manage finite resources as time unfolds.

This is a continuation of my other post which talked about how the
brain sorted the "picture" provided by the eyes.

Suppose I look at my small library of books. Each element of that
picture, such as the title of each book and its physical characteristics
begin as raw sensory visual data and are translated meaningfully by
the brain so that we see a coherent pattern, time after time. We can
also notice slight differences for instance when a book is missing.

I think the way the brain encodes the sensory data input into a
coherent pattern is very analogous to a definition and in English there
is a phrase like 'bring the picture into sharper definition' which is
connected to "focus" which gets back to which elements of a snapshot
of reality that our brain prioritizes and chooses to focus upon which
scientifically would be described by evolution of survival strategies.

The subject of physical capacities of the brain is investigated by
neurobiology. For instance it is known that an infant has twice as
many neurons as an adult, thus can form more neural patterns,
like learning more than one language. An adult has about 1/2 of
the number of neurons as an infant and has a much more difficult
time learning a second language. Which is why the old saying
"you can't teach an old dog new tricks" is true (applies to animals also).