[Fis] Formal Theory of Reflections

[Michel Petitjean]

Dear Krassimir and dear All,
Krassimir did a comprehensive review about relection theory.
This great work is extremely interesting, because it can be seen that,
behind the single word "reflection", there are several meanings,
depending on the context and depending on the culture.
We should realize that a similar situation occurs for the word
"information", so that exhibiting an unifying definition of
information remains a challenge.
About reflection in the sense of geometrical mirror images, the word
"chirality" appeared at the end of the 19th century to characterize an
object non identical to its mirror image.
Then the word "chirality" was reused by various authors outside the
Euclidean geometry context, thus generating confusion.
A mathematical definition of chirality based on goup theory exists,
and as such it works in any metric space:
> Chirality in metric spaces, In memoriam Michel Deza. Optim. Letters 2020,14[2],329--338. https://urldefense.com/v3/__https://doi.org/10.1007/s11590-017-1189-7__;!!D9dNQwwGXtA!WOGdRAumfARw1uezYCAgJWCDsXnMcqwjbfCxsa483KSy4kp90m3gKhk7ij56ri4OGPLOcFtJc4uhw-je-n_0diZ0MIkR$  (in OA).
It was further extended to Minkowski spacetime (e.g. see the OA paper
at https://urldefense.com/v3/__https://doi.org/10.3390/sym12040616__;!!D9dNQwwGXtA!WOGdRAumfARw1uezYCAgJWCDsXnMcqwjbfCxsa483KSy4kp90m3gKhk7ij56ri4OGPLOcFtJc4uhw-je-n_0diSqpb5o$ )
All that would not work without an unifying methematical definition of symmetry.
That latter was published in 2007 (free pdf https://urldefense.com/v3/__https://hal.science/hal-01552499__;!!D9dNQwwGXtA!WOGdRAumfARw1uezYCAgJWCDsXnMcqwjbfCxsa483KSy4kp90m3gKhk7ij56ri4OGPLOcFtJc4uhw-je-n_0dhb2t_Vg$ ).
But again, as "reflection" and "information", "symmetry" may be used
in a non mathematical sense.
It is why it is of crucial importance to be careful with words, and
when needed, to specify the context in which words are used.
An other example is "entropy", for which several definitions exists,
and which generate much confusion in the literature.
About "reflection theory", the review of Krassimir indeed helps.
Best regards,
Michel.
https://urldefense.com/v3/__http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html__;!!D9dNQwwGXtA!WOGdRAumfARw1uezYCAgJWCDsXnMcqwjbfCxsa483KSy4kp90m3gKhk7ij56ri4OGPLOcFtJc4uhw-je-n_0djRv5CBl$ 

Le jeu. 24 sept. 2026 à 20:30, Krassimir Markov <itheaiss at gmail.com> a écrit :
>
> Dear Colleagues,
> Let me inform you about publishing the second part of the Formal Theory of Reflections (FTR).
> Now it is available in full.
> Links to the papers with FTR:
> Part 1: https://urldefense.com/v3/__http://www.foibg.com/ijita/vol31/ijita31-03-p01.htm__;!!D9dNQwwGXtA!WOGdRAumfARw1uezYCAgJWCDsXnMcqwjbfCxsa483KSy4kp90m3gKhk7ij56ri4OGPLOcFtJc4uhw-je-n_0dvClqQem$ 
> Part 2: https://urldefense.com/v3/__http://www.foibg.com/ijita/vol31/ijita31-04-p01.htm__;!!D9dNQwwGXtA!WOGdRAumfARw1uezYCAgJWCDsXnMcqwjbfCxsa483KSy4kp90m3gKhk7ij56ri4OGPLOcFtJc4uhw-je-n_0dnwfN0K_$ 
> An abstract of the FTR is given below:
>
> Formal Theory of Reflections
> Abstract
> In the beginning, we traced the concept of reflection in extensive detail across Western philosophy from Plato and Aristotle through Stoicism, scholasticism, early modern philosophy, Kant, and phenomenology; across Buddhist, Daoist, Yogācāra, Advaita Vedānta, and Sāṃkhya thought; across Neoplatonic, Sufi, Hermetic, and Kabbalistic traditions; across the idealist–materialist debate; and across modern cognitive science, predictive processing, embodied cognition, neuroscience, and information theory.
> We then formalized reflection as the category Refl, with reflections as objects and contacts as morphisms, characterizing the identity morphism as reflection's null case and distinguishing it from self-reflection proper — a non-trivial endomorphism that may be either direct or transitive, depending on whether it factors through an intermediate reflection. We established Refl's basic properties, theorems, and cross-domain applications, extended it with functors, natural transformations, and a hierarchy of reflection levels, and introduced explicit temporal dynamics — throughout grounding the formalism in two recurring examples, a lake reflecting sunlight and a solar eclipse observed through several instruments.
> We raised the question: what distinguishes the subject from any other reflecting entity within the Formal Theory of Reflections? The answer is: a subject is an entity that compares — that relates an incoming reflection to reflections already held in memory, through an operator of its own, and acts on the result.
> We built this capacity for a single subject, grounding the once-informal notion of comparison in a precise, gestalt-derived, three-valued adequacy measure, and showing the resulting criterion to specialize, without revision, into the independently developed affective-cognitive cycle of the human case. Further we showed that because this capacity is internal to each subject, nothing can pass between two subjects directly — only through contacts mediated by a shared environment — and that the resulting chain of mutual adjustment, rather than any single transmission, is the right unit for understanding how subjects come to share, imperfectly but usably, one another's reconstructions. We showed that populations of such subjects, coordinated under shared rules rather than a common processor, can themselves satisfy an analogous criterion — collective memory, distributed comparison, distributed decision, connected into a cycle invariant under the replacement of any individual member — and illustrated this with a case, the ant colony, chosen precisely because it shares none of the human institutional apparatus that might otherwise make the claim seem merely metaphorical. We extended the same criterion across substrate, to the artificial subject, showing what it would structurally mean for an Infotron to compare, identifying a genuine disanalogy with the biological case that has no clean precedent anywhere and raising, without resolving, the question of whether this disanalogy marks a new level of reality (the Infotronic level) or an elaboration of the social one already established.
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