Toposcopy

Thank you for the excellent discussion on a central issue of epistemology.
The assertion that topology is a primitive ancestor to mathematics needs to
be clarified.

The assertion maintains, that animals possess an ability of spatial
orientation which they use intelligently. This ability is shown also by
human children, e.g. as they play hide-and-seek. The child hiding considers
the perspective from which the seeker will be seeing him, and hides behind
something that obstructs the view from that angle. This shows that the
child has a well-functioning set of algorithms which point out in a mental
map his position and the path of the seeker. The child has a knowledge of
places, in Greek "topos" and "logos", for "space" and "study".

As a parallel usage of the established word "topology" appears
inconvenient, one may speak of "toposcopy" when watching the places of
things. The child has a toposcopic knowledge of the world as it finds home
from a discovery around the garden. This ability predates its ability to
count.

The ability to be oriented in space predates the ability to build abstract
concepts. Animals remain at a level of intellectual capacity that allows
them to navigate their surroundings and match place and quality attributes,
that is: animals know how to match what and where. Children acquire during
maturing the ability to recognise the idea of a thing behind the perception
of the thing. Then they learn to distinguish among ideas that represent
alike objects. The next step is to be able to assign the fingers of the
hand to the ideas such distinguished. Mathematics start there.

What children and animals have and use before they learn to abstract into
enumerable mental creations is a faculty of no small complexity. They
create an inner map, in which they know their position. They also know the
position of an attractor, be it food, entertaintment or partner. The
toposcopic level of brain functions determines the configuration of a
spatial map and furnishes it with objects, movables and stables, and the
position of the own perspective (the ego).

This archaic, instinctive, pre-mathematical level of thinking must have its
rules, otherwise it would not function. These rules must be simple,
self-evident and applicable in all fields of Physics and Chemistry, where
life is possible.  The rules are detectable, because they root in logic and
reason. The rules may be hard to detect, because, as Wittgenstein puts it:
one cannot see the eye one looks with, fish do not see the water. We
function by these rules and are such in an uneasy position questioning our
fundamental axioms, investigating the self-evident.

The rules have to do with places and objects in places. Now we imagine a
lot of things and let them occupy places. It is immediately obvious that
this is a complicated task if one orders more than a few objects according
to several, different aspects.

We introduce the terms: collection, ordered collection, well-ordered and
extremely well ordered. As a collection we take the natural numbers, in
their form of a+b=c. This set is ordered, as its elements can be compared
to each other and a sequence among the elements can be established. We call
the collection well-ordered, if every aspect that can create a sequence
among the elements is in usage, determining the places of elements in
sequences. A well-ordered collection can not be globally and locally stable
at the same time. In most parts and at most times, it is in a quasi-stable
state. The instabilities coming from contradictions among the implications
of differing orders regarding the position of elements will appear in many
forms of discontinuities. We call the collection extremely well-ordered, if
the discontinuities, which appear as consequence of praemisses which are no
more compatible to each other, in their turn cause such alterations in the
positions of the elements that henceforth the praemisses are again
compatible to each other. The extremely well-ordered collection maintains a
loop of consequences becoming causes while changes in spatial
configurations take place. In the well-ordered collection there is a
continuous conflict, out of which loops that maintain stability can evolve.

The mechanism is easy to recreate on one's own computer. Nothing more than
a few hours of programming is required to understand and to be able to use
the toposcope. Its main ideas are known under "cyclic permutations". It is
important to visualise that elements change places during a reorder. The
movement between "previously correct, now behind me", "presently here, not
yet all stable" and "correct in future, not yet there" has many gradations
and many places. Patterns evolve by themselves, as properties of natural
numbers.

There is a simple set of numeric facts that build the backbone of spatial
orientation. The archaic knowledge shared by animals and children is based
on a simple set of algorithms. These algorithms predetermine the connection
between where and what. The toposcopic brain utilises the numeric facts,
like the liver utilises the chemical facts.

The layer of interpretations of the world that is a pre-human, animal,
instinctive knowledge about spatial orientation needs no learning, because
it is based on facts. The facts are not, where it will condense and what it
will look like, but rather the facts are that there will be a region where
it will condense and it will have a specific property to it. The patterns
of movements of elements during changes in order in a well-ordered
collection create a basic sceleton of thinking. To see the patterns here
referred to, it is necessary to order a collection and then order it some
more until it becomes well-ordered, and watch the conflicts that are
immanent to order, namely its alternatives and its background.  This is
simple, archaic and instructive.
_______________________________________________
Fis mailing list
Fis@listas.unizar.es
http://listas.unizar.es/cgi-bin/mailman/listinfo/fis

Reply via email to