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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