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The topology of knowledge: In this application, topological spaces are used to model the structure of knowledge, where the open sets correspond to coherent bodies of knowledge and the closure operation represents the process of inference.

I've heard that we use topology to model knowledge. I've read that topological spaces are used to model the structure of knowledge, and we use open sets to represent coherent bodies of knowledge and closure operation to represent the process of inference. However, I am not very familiar with the concepts of "open sets" and "closure operation". Could you expand on this and give a more detailed explanation?

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  • You should probably learn the topology part first, because works on topology of knowledge that I've seen presuppose it as a prerequisite. For a crash course, Wikipedia has articles on open sets and closure. These are not topics suitable for this SE.
    – Conifold
    Commented Feb 11, 2023 at 22:37

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This is an interesting question. I am pretty knowledgeable in point set topology, and I have recently read a lot of Alexander Dugin, he is very a controversial russian genius, but i find his work fascinating and he addresses your question in one of his "Noomachia" books dedicated to Ancient greek thinking (i think it's like volume 16 or something).

There are 3 epistemologies / theories of knowledge. The first one is Appollonic (in an almost Nietzschiean sense, but not quite) which is exemplified in Plato and Christianity. This is the worldview for which only the eternal / the idea / the form / the logos / the god is real, and everything else is an illusory shadow. This knowledge is modelled by Pythagorean number theory and sacred geometry (not geometry as measuring land surveying, but rather as Platonic solids and Euclidean axioms).

On the other extreme there is the logos of Cybele, which is the great subterranean mother. Out of this we get post-modernism, atomic physics and mathematically this knowledge is best modelled by chaos theory, fractals and the bizarre findings of Benois Mandelbrot.

In between the light Appollonic heaven and the dark Cybelene hell there is the fascinating domain of Dyonisis. This is essentially our phenomenal world, it is the world as a phenomenal field / horizon. Aristotle was the quintessential Dionisian thinker in Ancient Greece, and in modern times his doctrines were enriched and revived in phenomenology, Brentano began his intellectual career as a re-interpreter of Aristotle.

Topology is mathematical phenomenology. It is math of the phenomenal field. Open sets are the building blocks of topological spaces, and they are intentional acts. An open set is sort of between the finite and the infinite, it has 2 horizons, which it never reaches. Likewise, intentional acts have 2 horizons, one finite, and one infinite, they can never give you access to the transcendental, and they can not be reduced to mere finite physicality and biology.

Topology = mathematical model of the here and now, between the spiritual eternal realm above and the physical realm of finite matter. Topology is mathematics that exists between platonism and materialism.

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Well, topology is a big word for a model most of us would just call a map. In computer science, knowledge, which is defined a little bit differently than in philosophy, is structured with representations called graphs. Graphs are topological because they are concerned with relationships and metrics that are imputed to describe a space, which is also defined a little differently than in vernacular English.

In fact, graphs are used all the time to model human understanding in artificial intelligence, and are of great interest in modeling human knowledge because knowledge about the topology of graphs has philosophical implications on what can and cannot be known. Famously, the traveling salesman problem is described in the computational term of NP-hardness and is central in questions of computational complexity. These days, philosophers of mind take a big interest in Turing machines, computational theory and complexity, and computer languages because they have a number of commonalities with human brains, thought, and natural language. In fact, Curry-Howard-Lambek claims there is a fundamental similarity between logic, computation, and category theory which itself is expressed in graph-theoretic language.

If you want to know how graphs, and thus topologies, are used to model knowledge, any book on abstract data types will be a great introduction. Today, with the academic discipline of experimental philosophy becoming more recognized and popular, understanding just how data structures of computers mimic or recreate human knowledge is becoming more and more developed. In fact, one of the popular positions in cognitive science and philosophy of mind is that of the Computational Theory of the Mind (SEP).

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