Dear Karl,
Thanks for your wise
comments.
You wrote: "The session so far has raised the
points: meta-communication, subject-matter, order, spaces.
a.) Meta-communication
Gordana’s summary explicates the need to have a system of references
that FIS can use to discuss whatever it wishes to discuss, be it the
equivalence between energy and information or the concept of space in the human
brain. Whatever the personal background, interests or intellectual creations of
the members of FIS, we each have been taught addition, multiplication, division
and the like. We also know how to read a map and remember well where we had put
a thing as we are going to retrieve it. When discussing the intricate,
philosophical points which are common to all formulations of this session, it
may be helpful to use such words and procedures that are well-known to each one
of us, while describing what we do while we use topology".
I agree with you. I will try to follow this rule. …however, read my response
to your fourth
point…
b.) Subject-matter
Topology is managed by much older structures of the central
nervous system than those that manage speech, counting, abstract ideas. Animals
and small children remember their way to food and other attractions. Children
discover and use topology far before they can count. Topology is a primitive
ancestor to mathematics; its ideas and methods are archaic and may appear as
lacking in refinement and intelligence.
This time, of course, I cannot
agree. Topology is not a primitive
ancestor that stands just for the older
brain structures, and is not tenable that children discover topology far
before they can do other activities: nobody
knows that, and the literature is controversial. Rather, topology is a sort of
meta-scientific
tool: because its abstractness and ability
to describe very general features of structures and objects, it allows the
assessment
of almost all the physical and biological phenomena. The trick is just to find
the proper way to
transfer such matematical concepts from an abstract phase space to a real,
experimentally assessable one, the one where biological/physical activities
take place.
Look at my very brief movie on Youtube (just one minute!):
https://www.youtube.com/watch?v=oxfqraR1bIg
If you change the described 2D circle and the 3D sphere with other structures
(for example, the 2D flattened cortex and the 3D whole brain), the trick is
easier to understand.
Therefore, topology is able
to give novel insights in countless contexts, from pre-Big Bang scenarios, to
quantum entanglement, from biological gauge fields, to semantics, and, of
course, to brain activity. The standpoint
of topology, e.g., mappings and projections between levels equipped with
different dimensions (either spatial, or temporal, or abstract dimensions), is
a tenet that can be used in the assessment of every scientific activity.
c.) Order
There is no need to discuss whether Nature is well-ordered or not.
Our brain is surely extremely well ordered, otherwise we had seizures, tics,
disintegrative features. In discussing topology we can make use of the
condition that everything we investigate is extremely well ordered. We may not
be able to understand Nature, but we may get an idea about how our brain
functions, in its capacity as an extremely well ordered system. We can make a
half-step towards modelling artificial intelligence by understanding at first,
how artificial instincts, and their conflicts, can be modelled. Animals
apparently utilise a different layer of reality of the world while building up
their orientation in it to that which humans perceive as important. The path of
understanding how primitive instincts work begins with a half-step of dumbing
down. It is no more interesting, how many they are, now we only look at where
it is relative to how it appears, compared with the others.
The differences in complexity and
in building up of perceptions in different animals can be easily framed in a
topological context that explains them in terms of different (functional, not
spatial!) dimensions. The higher the
number of dimensions, the higher the complexity and the stored
information. Primitive istincts, in a
topological framework, are not very different from higher brain activities: the
only difference lies in the dimension we are evaluating them. We
"anthropocentrically" take into account just
the dimensions we prefer: therefore, looking by a given level, we believe that
the others are less interesting. It is
not true: all the levels display the same content, even if with different
“quantity”
of information. We see things from our
standpoint (we can say: from a single topological dimension).
d.) Spaces
Out of sequences, planes naturally evolve. Whether out of the
planes spaces can be constructed, depends on the kinds of planes and of common
axes. Now the natural numbers come in handy, as we can demonstrate to each
other on natural numbers, how in a well-ordered collection the actual mechanism
of place changes creates by itself two rectangular, Euclidean, spaces. These
can be merged into one common space, but in that, there are four variants of
every certainty coming from the position within the sequence. Furthermore, all
these spaces are transcended by two planes. The discussion about an oriented
entity in a space of n dimensions can be given a frame, placed into a context
that is neutral and shared as a common knowledge by all members of FIS.
A change of “frame” is
required. The “neutral” framework of an
“oriented entity in a space of n-dimensions” is not valid, if you just look at
the same "entity" from just one dimension higher.
In the same vein, in touch with, e.g., Spencer, we can say that
biological evolution is just a path towards an higher number of dimensions
(see, e.g., the case of the neural plate quoted by another FIS member): in
this case, we take into account not spatial dimensions, but complexity
dimensions. The higher the dimensions,
the higher the complexity. The basic ingredients
are the same for all the levels, e.g., a bit of matter and of energy
(interchangeable), but evolution takes place (we can rather improperly use the
classical word “emerges”) when these few ingredients are located in higher
levels of complexity, living rise to a local increase of information. The
original 2D shadow of a cat, when evaluated
in 3D, becomes a true cat! This is
topology.
Thanks a lot for your attention!
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