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.
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