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I am going to mumble about couple of things I really don't know and I am going to ask for guidance and further reading material.

There is a field of mathematics called topology which studies continuous deformations of objects into each other and what stays the same during these continuous deformations.

Do you know any formal or informal application of topology to philosophical ideas? For example, take history of eros. Do we have a continuous change throughout or collection of discrete changes brought about by revolutions? Can any similar formal language be built to talk about seemingly non-mathematical concepts? How the understanding of a concept is deformed through history in our collective minds?

Now take on all the above questions and I would be glad if you shared your own insights, relevant thinkers, reading materials etc.

Byung-Chul Han has a book called Topology of Violence in which he actually tracks our understanding of violence and how we happened to internalize it in stages etc. But the name is mostly an inspiration, rather than application of topological methods. I wanted to look out for more formal applications but I am lost.

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    I doubt very much that formal methods in topology or indeed from any part of mathematics applies in such fields. Though Lacan, from what I recall, has used topological ideas in his theories. I think it's safe to say they're best seen as a metaphor/inspiration. – Mozibur Ullah Feb 9 '18 at 14:39
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    Karatani Kojin, just as Lacan used Borromean rings to develop some ideas. Closely related appear to be some of Rene Thom's writings, exploring 'catastrophe theory'; he has published in 1970 a paper "Topologie et linguistique". – sand1 Feb 9 '18 at 22:08
  • "Spacing Freud: Space and Place in Psychoanalytic Theory" by Nicholas Dion. PDF tspace.library.utoronto.ca/bitstream/1807/33977/1/… – Gordon Feb 10 '18 at 17:43
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Peirce was one of the first to invoke topology in his metaphysics. The notion of continuum and continuity is central to his account of the basic philosophical categories, and he was even led to study three-valued logic by topological considerations (status of points on the boundaries between regions). See Peirce’s Topological Concepts by Havenel:

"Peirce thinks that the study of topology could help to solve many philosophical questions. For example, the idea of continuity is of prime importance for Peirce’s synechism, and topology is “what the philosopher must study who seeks to learn anything about continuity from geometry” (NEM 3.105). In Peirce’s reasonings concerning the nature of time and space, topological concepts are essentials. Moreover, in his logic of existential graphs and in his cosmology, the influence of topological concepts is very apparent."

Introduction: The Becoming Topological of Culture by Lury et al. is a recent survey of analogical application of topological ideas to cultural theory:

"The influence of topology in social and cultural theory in recent decades is immense. Topological ideas have been a significant source of inspiration across many social science disciplines, including philosophy, sociology, political science, psychology, anthropology, geography and economics. Topological ideas have fed into and transformed multiple specific fields of study. So, for example, as Marres (2012) observes, in the field of social studies of technology, topological ideas have helped dismantle the view that technology and society occupy different domains, contributing instead to the concept of a heterogeneous ‘assemblage’, which is heterogeneously composed of social, technical and natural enti ties (Latour, 1987)...

In all these approaches it is relationality that is important, and topological concepts have enabled social theory to move beyond a reliance on the mechanical and organic to include transductive and transitive modes of relating (Simondon, 1992), and helped turn space and time from ‘a priori’ into ‘a posteriori’ categories (Lash, 2009). But what is proposed in this introduction is not simply the transposition of topological ideas onto the field of culture. Instead we are interested in teasing out an epochal transformation in the intersection between the form and content of cultural expression... our proposal is that topology is now emergent in the practices of ordering, modelling, networking and mapping that co-constitute culture, technology and science.".

For a critical philosophical discussion of this "topological turn", anticipated by Lacan and Badiou, see On Topology by Phillips, who is advocating a different, "non-quantitative", "inexact", conception of topological in culture, one inspired by Heidegger:

"This article questions the novelty of this ‘becoming topological of culture’ and digs into a deeper historicity in order to identify the trends that may be said to support the development of topology in the current situation... For instance, mathematical objects, despite intersections, seem relatively distinct from those of the social sciences. But the idea of a topological turn in social theory suggests at the very least that the latter, the objects of political and cultural theory, say, can be addressed as if they were of the same kind as those of the former.

[...] I think, though, that the high level of abstraction required for the construction of a topological space might give the impression that some kind of labile inexactitude operates there too – which I believe is false. You don’t really need mathematics for labile inexactitude. The capacity to absorb a vocabulary of topological transformation (mapping, open sets, neighbourhoods, homologies and so on) into an already existing vocabulary derived from the structuralism of the turn of the last century might better be examined by taking the historicity of concepts as a guide."

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You asked for reflections too, so I hope you don't mind this reflection on my part. My own feeling is that there are no breaks in history, but there is "bad" history. Sometimes the record is simply lost, sometimes it is suppressed on purpose, and sometimes the record is neglected (new is better, the old is forgotten).

As time proceeds, we generate "unworthy fragments" what I mean by this is incomplete fragments from the past. With postmodernism the idea may be to celebrate these fragments rather than to try to complete their proper history so far as it is possible. This postmodernistic sloppiness and laziness is mankind letting itself be fooled in my opinion. The more good history we have ( as good as we can make it, i.e. the more hard work in an admittedly messy and difficult endeavor) the more seamless it becomes.

We humans are mortal, our individual lives come to an end, and in this sense we are discrete, so we tend to think of "breaks" rather than the continuum of history. Maybe a topology could be used in a suggestive or heuristic manner, with due caution that it will likely lead us astray if we adhere to it too tightly. I hope I have understood your question.

Since I have posted a paper above in the Comments which may deal at least in some way with your specific question, I will take the liberty of posting an essay that I have found interesting on the subject of history, and teaching history. "Sartre On Our Responsibility For Dead Lives: Implications For Teaching History" https://www.bu.edu/wcp/Papers/Hist/HistGord.htm

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